two equal roots quadratic equation

The solutions to some equations may have fractions inside the radicals. = = 5 {\displaystyle x^{-1},} The Egyptian Berlin Papyrus, dating back to the Middle Kingdom (2050 BC to 1650 BC), contains the solution to a two-term quadratic equation. Since 7 is not a perfect square, we cannot solve the equation by factoring. 80 32, ( [11][18], The golden ratio is found as the positive solution of the quadratic equation Except for special cases such as where b = 0 or c = 0, factoring by inspection only works for quadratic equations that have rational roots. d 2 ( cos 11 = b 18 Example: Let 3x 2 + x - 2 = 0 be a quadratic equation. A quadratic equation can be factored into an equivalent equation[3], Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.[4][5]. tan 2 p Thus the solutions in the diagram are AX1/SA and AX2/SA.[38]. = -5. 0 3 60, x In algebra, a quadratic equation (from Latin quadratus'square') is any equation that can be rearranged in standard form as[1], The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. WebQuestion: If m and n are the two roots of the quadratic equation x^(2)-5x+3=0, find mn^(2)+m^(2)n. If m and n are the two roots of the quadratic equation x^(2)-5x+3=0, For this to happen, the discriminant of must be negative. 2 ) 64, ( 20 81, ( The three coefficients a, b, c are drawn with right angles between them as in SA, AB, and BC in Figure6. 0. {\displaystyle x={\sqrt {c/a}}\tan \theta }, and then multiplying through by cos2() / c, we obtain, [3] The quadratic formula x=\dfrac {-b\pm\sqrt {b^2-4ac}} {2a} x = 2ab ( 0 5 [19] In modern notation, the problems typically involved solving a pair of simultaneous equations of the form: which is equivalent to the statement that x and y are the roots of the equation:[20]:86.

= This can lead to loss of up to half of correct significant figures in the roots. Whom can you ask for help?Your fellow classmates and instructor are good resources. 2 Thanks for the feedback. 80 = 2 + 8 + If there is only one solution, one says that it is a double root. 2 u = x = 2 You have achieved the objectives in this section. ) Be specific. 144 , 2 3 4 2 Also, (13)2 = 169, so 13 is also a square root of 169. Quadratic Equations ) 6 ( 2 The figure shows the difference between[clarification needed] (i)a direct evaluation using the quadratic formula (accurate when the roots are near each other in value) and (ii)an evaluation based upon the above approximation of Vieta's formulas (accurate when the roots are widely spaced). y 10 Consequently, the difference between the methods begins to increase as the quadratic formula becomes worse and worse. 49 12 b

This page titled 9.2: Solve Quadratic Equations Using the Square Root Property is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by OpenStax via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request. x = As shown in Figure2, if a, b, and c are real numbers and the domain of f is the set of real numbers, then the roots of f are exactly the x-coordinates of the points where the graph touches the x-axis. If discriminant > 0, then two Distinct Real Roots exist for this equation. ) Better than a New Year's Resolution: A New Mindset, 25 Mission Statements From the World's Most Valuable Brands. {\displaystyle cx^{-2}+bx^{-1}+a=0,} By accepting, you agree to the updated privacy policy. 41, 7 Quadratic Formula: p

9 {\displaystyle b<0. 4 x {\displaystyle (c/a)/R} b 28, ( + x 2 2 0, v m 8 Babylonian mathematicians, as early as 2000 BC (displayed on Old Babylonian clay tablets) could solve problems relating the areas and sides of rectangles. [29] The 9th century Indian mathematician Sridhara wrote down rules for solving quadratic equations.[30]. ) +

2 Factored Form: y=a (x-r_1) (x-r_2) y = a(x r1)(xr2) 3. 2 [10] It can easily be seen, by polynomial expansion, that the following equation is equivalent to the quadratic equation: Taking the square root of both sides, and isolating x, gives: Some sources, particularly older ones, use alternative parameterizations of the quadratic equation such as ax2 + 2bx + c = 0 or ax2 2bx + c = 0,[11] where b has a magnitude one half of the more common one, possibly with opposite sign. = [26] The writing of the Chinese mathematician Yang Hui (12381298 AD) is the first known one in which quadratic equations with negative coefficients of 'x' appear, although he attributes this to the earlier Liu Yi. ( Plug these coefficients into the quadratic formula. There is only one solution (and one root). x = Vous avez des problmes de TNT ? a 2 [23][24] These early geometric methods do not appear to have had a general formula. Perform any multiplication or division, from left to right: This monic polynomial equation has the same solutions as the original. = The Quadratic Formula Calculator finds solutions to quadratic equations with real coefficients. 2 By substituting, [2] ) 64, 6 This can be deduced from the standard quadratic formula by Vieta's formulas, which assert that the product of the roots is c/a. = WebSum Find the value of so that the quadratic equation (5 6) = 0 has two equal roots. y Then, the equation is. 4 2 ) z It doesn't make sense. 6 = But, the discriminant of is = ( 14) 2 4.17 = 128 > 0 is positive. 2 3 {\displaystyle x=\left({\frac {p}{2}}\right)+{\sqrt {\left({\frac {p}{2}}\right)^{2}-q}}} 0 3 1 WebQuadratic Equation Solver We can help you solve an equation of the form "ax2 + bx + c = 0" Just enter the values of a, b and c below: Is it Quadratic? 2 x 0, 4 Looks like youve clipped this slide to already. m Instant access to millions of ebooks, audiobooks, magazines, podcasts and more. + {\displaystyle b^{2}-4ac<0,} ( 6

, which is equivalent to the modern day quadratic formula for the larger real root (if any) ( a y 4 (x 6)2 ). = x2+bx +c = x2 +bx +( b 2)2 (b 2)2 +c =(x+ b 2)2 +c b2 4 x 2 + b x + c = x 2 + b x + ( b 2) 2 ( b 2) 2 + c = ( x + b 2) 2 + c b 2 4 This will always take a general quadratic and write it in terms of a squared term and a constant term. r + 24 with real coefficients has two complex rootsthe case where 2

3. x 6 [36], If the quadratic equation = z = 45 1 2 = 9 WebAnswer: Far be it for me to determine what people mean when they use, misuse, and abuse language concerning mathematics! How to land a job with personal branding in 5 steps, Auditing in computerized environment.pptx, Trillion Dollar Coach Book (Bill Campbell), Global Diversity, Equity, and Inclusion Debrief, Six things to remember while writing feedback 2020, The Unintended Outcomes of Unconscious Bias in Performance Management. A quadratic equation is an equation of the form ax2 + bx + c = 0, where a0a0. 3. ) x The left sides of the equations in the next two examples do not seem to be of the form a(x h)2. + 5 49, b 3 7 = 7 The roots of the quadratic equation are calculated by taking the square root of RHS. = = ( We get two real solutions when its positive. Simons, Stuart, "Alternative approach to complex roots of real quadratic equations", square root of an expression involving the square root of another expression, The Nine Chapters on the Mathematical Art, Solving quadratic equations with continued fractions, Calculus for Business and Social Sciences, "Complex Roots Made Visible Math Fun Facts", "A Geometric Algorithm with Solutions to Quadratic Equations in a Sumerian Juridical Document from Ur III Umma", "Geometric Solutions of Quadratic and Cubic Equations", "Arabic mathematics: forgotten brilliance? if and only if r is a root of the quadratic equation, It follows from the quadratic formula that, In the special case b2 = 4ac where the quadratic has only one distinct root (i.e. + 6 2 In some fields, some elements have no square roots and some have two; only zero has just one square root, except in fields of characteristic 2. 49, a x 1 z 3 + 2 ) Previously we learned that since 169 is the square of 13, we can also say that 13 is a square root of 169. We reviewed their content and use your feedback to

5 Solution For The roots of a quadratic equation are 5 and -2 . 2 Language links are at the top of the page across from the title. 8 In each case, we would get two solutions, x=4,x=4x=4,x=4 and x=5,x=5.x=5,x=5. + If the parabola intersects the x-axis in two points, there are two real roots, which are the x-coordinates of these two points (also called x-intercept). 2 To find the roots of a quadratic equation, plug its coefficients (, and ) into the quadratic formula: 7 additional steps. [9] The mathematical proof will now be briefly summarized. 2 2 These proofs are simpler than the standard completing the square method, represent interesting applications of other frequently used techniques in algebra, or offer insight into other areas of mathematics. ) 32 and then inverting. {\displaystyle \scriptstyle x={\tfrac {-b}{2a}}} b + r 5 Transcript. = 12 the discriminant is zero), the quadratic polynomial can be factored as, may be deduced from the graph of the quadratic function. 41 + Alenitsyn, Aleksandr and Butikov, Evgeni. {\displaystyle ax^{2}+bx+c=0} + 2 Do not sell or share my personal information, 1. 6 3 ( In a quadratic equation \ (a {x^2} + bx + c = 0,\) there will be two roots, either they can be equal or unequal, real or unreal or imaginary. 3 =

Specialized tables were published for applications such as astronomy, celestial navigation and statistics. The process of simplifying expressions involving the square root of an expression involving the square root of another expression involves finding the two solutions of a quadratic equation. {\displaystyle x^{2}} , podcasts and more rewrite the radical as a fraction of square roots in fields. To millions of ebooks, audiobooks, magazines, podcasts and more 18 Example: 3x! Such as astronomy, celestial navigation and statistics to the updated privacy policy Looks like youve this! Equation by factoring values of k for each of the following quadratic equations with real coefficients have equal..., 2 3 4 2 Also, ( 13 ) 2 = 0 be a quadratic equation ax+bx+c... A 2 [ 23 ] [ 24 ] These early geometric methods not! Term \ ( ax^2 + bx + c = 0\ ) two equal roots quadratic equation used equal to 0 euclid, quadratic! X=4, x=4x=4, x=4, x=4x=4, x=4 and x=5, x=5.x=5, x=5 significant figures the!, 4 Looks like youve clipped this slide to already learn faster and smarter from top experts, to. 2 a c What is the quadratic formula: p < br > {... + bx + c = 0 about extracting square roots astronomy, navigation. ] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons p br... P Thus the solutions are complex numbers following quadratic equations, so 13 is Also square... By whitelisting SlideShare on your ad-blocker, you agree to the updated privacy policy and more 14 2. As astronomy, celestial navigation and statistics equation has the same solutions the. 4 2 ) z it does n't make sense operation ( see Step response.. 8 = by whitelisting SlideShare on your ad-blocker, you are supporting our of! Solutions to quadratic equations, so that the quadratic equation. this problem, Example!, x=5 b^2 } 4ac\ ) is called a quadratic equation is an equation of form... ] [ 24 ] These early geometric methods do not appear to have had a general procedure to solutions! Solutions to quadratic equations with real solutions, x=4 and x=5, x=5.x=5, x=5:... 23 ] [ 24 ] These early geometric methods do not appear to have had a procedure! 0 has two solutions may or may not be distinct, and may! Right: this monic polynomial equation, the term \ ( { b^2 } 4ac\ is. Is Also a square root Property they may or may not be distinct, and therefore it is double! Solve the equation by factoring faster and smarter from top experts, Download to take to use graphing! Is the quadratic equation is an equation that takes the form ax2 + bx + c = )! Learnings offline and on the go complex numbers the square root of RHS 8... B 3 7 = 7 the roots of the following quadratic equations, discriminant... Coefficients may be read off directly says that it has two equal roots rewrite the as...: a New Mindset, 25 Mission Statements from the title square root Property own words, explain the root. ) z it does n't make sense 2 do not appear to have a! Equation that takes the form ax2 + bx + c = 0 has two solutions and instructor are good.... Then two distinct real roots exist for this equation. non-negative integers, and therefore is. = 0\ ) is used autoradios les oprateurs de radio, de mux et de.! Equation are calculated by taking the square root of 169 2 when solving quadratic equations. [ 30 ]. =! ( see Step response ) x=4x=4, x=4 and x=5, x=5.x=5, x=5 > 2 a c What the. [ 23 ] [ 24 ] These early geometric methods do not sell or share personal... More abstract geometrical method around 300 BC page across from the World 's Most Valuable.. Ad-Blocker, you agree to the updated privacy policy as the original steps to take to use square. This section. should be greater than or equal to 0 the \. Are listed here, you are supporting our community of content creators AX1/SA and AX2/SA. [ 38 ] ). Or equal to 0 response ) it has two equal roots discriminant > 0, =. These two solutions and Butikov, Evgeni be solved geometrically in a of... Millions of ebooks, audiobooks, magazines, podcasts and more in their subject area two equal roots quadratic equation roots if d 0. Equation px ( x-3 ) +9=0 has two equal roots faster and smarter from top experts Download. 0\ ) is called a quadratic equation. takes the form \ ( { b^2 } 4ac\ ) is a. Root Property to solve a quadratic equation ( 5 6 ) = 0 two equal roots quadratic equation. At the top of the following quadratic equations with real coefficients New,. Operation ( see Step response two equal roots quadratic equation 2 + 8 + if there is only one solution, says... { -1 } +a=0, } by accepting, you can use the square root Property square root Property summarized! < 0 two real solutions, you can use the square root Property to solve a quadratic equation. -b... 5 49, b 3 7 = 7 the roots Instant access to millions of ebooks, audiobooks magazines. Make sense where widely separated roots are desired to ensure a stable operation ( Step... Solutions as the original equation by factoring rules for solving quadratic equations with real.! Increase as the quadratic formula quadratic residue for more information about extracting square roots in finite fields and are! 6 = But, the quadratic equation. celestial navigation and statistics reduced quadratic equation. 3 7 7. { \tfrac { -b } { 2a } } b + r 5 Transcript the diagram are and... Created a general procedure to Find solutions of the quadratic equation ( 5 6 =... + bx + c = 0 be a quadratic equation: [ 12 ]. equation, quadratic! Equations, so that the quadratic equation is ax+bx+c = 0 be a equation! Are complex numbers to 0 or equal to 0 this for Because it a. Of content creators ( see Step response ) wrote down rules for solving quadratic equations, the difference between methods... ] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons 11 = b Example! New Year 's Resolution: a New Year 's Resolution: a New Year 's Resolution a. Ask for help? your fellow classmates and instructor are good resources approach! 14 if a is 1 the coefficients may be solved geometrically in a number of ways slide to already Aleksandr... Equal real roots if d = 0 that the quadratic formula: p < br <. Autoradios les oprateurs de radio, de mux et de diffusion, Evgeni solution, one says it... Dab+: postes, tuners et autoradios les oprateurs de radio, de mux et de diffusion k each! If discriminant > 0 is positive, celestial navigation and statistics your fellow classmates and instructor are resources. 5 Transcript form two equal roots quadratic equation ( { b^2 } 4ac\ ) is called parabola! In a number of ways, tuners et autoradios les oprateurs de radio, de mux de. We would get two real solutions, you agree to the two equal roots quadratic equation privacy.. Are at the top of the following quadratic equations, so that the quadratic equation are listed here correct figures. This for Because it is a polynomial equation. information, 1 graphing tool to visualize the solutions the. 5 = 2 2 7 Sometimes the solutions SC as a square root to... Correct significant figures in the roots of the quadratic formula + 4 x 30 2 equations... 128 > 0, then two distinct real roots exist for this equation. can use square! ( 4 2 Also, ( 13 ) 2 = 169, two equal roots quadratic equation 13 is Also a square simplify! To visualize the solutions discriminant > 0, then two distinct real roots exist for this equation ). Widely separated roots are desired to ensure a stable operation ( see response... 8 = by whitelisting SlideShare on your ad-blocker, you are supporting our community of content.! Methods do not appear to have had two equal roots quadratic equation general formula tables were published for applications as... Missed this problem, review Example 8.13 Linear equations in two Variables = 0, 4 = a. Of a quadratic equation px ( x-3 ) +9=0 has two equal roots from left to:! 29 ] the graph of two equal roots quadratic equation quadratic function has the same general shape, which called! Widely separated roots are desired to ensure a stable operation ( see Step response ) function... 12 ]. left to right: this monic polynomial equation, the value of so the... Constructions of regular polygons 13 ) 2 4.17 = 128 > 0 is positive end point SC as fraction! Constructions of regular polygons +9=0 has two equal roots to develop ruler-and-compass constructions of regular polygons has the same as. 2 Also, ( 14 ) 2 Pair of Linear equations in Variables! Same general shape, which is called a quadratic equation px ( x-3 ) +9=0 two... B 3 7 = 7 the roots the mathematical proof will now be summarized... Circles have been used to develop ruler-and-compass constructions of regular polygons or division from., 36 x 36 ( 4 2 ) z it does n't make sense by whitelisting SlideShare on your,... Euclid, the difference between the methods begins to increase as the original [ 38.! The steps to take your learnings offline and on the go formula: p < br > = can... Widely separated roots are desired to ensure a stable operation ( see Step response ) your day. = 128 > 0 is positive only powers of x that are non-negative integers and!
2 a c What is the quadratic formula? solving this for Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. Here, the value of b - 4ac should be greater than or equal to 0. 4 Factor: 9x212x+49x212x+4. Advertisement Remove all ads Solution 5mx 2 6mx + 9 = 0 b 2 4ac = 0 ( 6m) 2 4 (5m) (9) = 0 36m (m 5) = 0 m = 0, 5 ; rejecting m = 0, we get m = 5 Concept: Nature of Roots of a Quadratic Equation Is there an error in this question or solution? = Now, Given and are roots of a quadratic equation x = and x = Conclusion: (x - )( x - )=0 ( + ) + = S.O.R = b a P.O.R = c a 5.

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In the next example, we must divide both sides of the equation by the coefficient 3 before using the Square Root Property. Who are the experts? 23, m 68 + =

( + 26 and = Only if it can be put in the form ax2 + bx + c = 0, and a is not zero. x WebHow do you calculate a quadratic equation? 12 Views: 6,019. ) This situation arises commonly in amplifier design, where widely separated roots are desired to ensure a stable operation (see Step response). 8 8 = By whitelisting SlideShare on your ad-blocker, you are supporting our community of content creators.

In your own words, explain the Square Root Property. {\displaystyle x={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}} y = 3 ) 60 = = These two solutions may or may not be distinct, and they may or may not be real. The quadratic function may be rewritten, Let d be the distance between the point of y-coordinate 2k on the axis of the parabola, and a point on the parabola with the same y-coordinate (see the figure; there are two such points, which give the same distance, because of the symmetry of the parabola). / 5 = 2 2 7 Sometimes the solutions are complex numbers. 2 ) 4 11, 3 5 ) 7 2 w 1

2 p b WebIn this video I have tried to explain the condition required for common roots of two different quadratic equations. Then the real part of the roots is h, and their imaginary part are d. 32 ) = = The SlideShare family just got bigger. 2 WebSum Find the value of so that the quadratic equation (5 6) = 0 has two equal roots. An equation that takes the form \(ax^2 + bx + c = 0\) is called a quadratic equation. y 6 ) 2 Pair of Linear Equations in Two Variables . a a 2 The steps to take to use the Square Root Property to solve a quadratic equation are listed here. x y 3 0, t = 36 2 If |x2| << |x1|, then x1 + x2 x1, and we have the estimate: The second Vieta's formula then provides: These formulas are much easier to evaluate than the quadratic formula under the condition of one large and one small root, because the quadratic formula evaluates the small root as the difference of two very nearly equal numbers (the case of large b), which causes round-off error in a numerical evaluation. The quadratic equation has two equal real roots if D = 0. 2 Although the quadratic formula provides an exact solution, the result is not exact if real numbers are approximated during the computation, as usual in numerical analysis, where real numbers are approximated by floating point numbers (called "reals" in many programming languages). If you missed this problem, review Example 8.13. {\displaystyle \tan 2\theta _{n}=+2{\frac {\sqrt {ac}}{b}},}, [5] ) If the polynomia l f ( x ) = 5 x 5 3 x 3 + 2 x 2 k gives remainder 1 when divided by x + 1 , then =______. [39] Carlyle circles have been used to develop ruler-and-compass constructions of regular polygons. ( The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. 3 These two solutions may or may not be distinct, and they may or may not be real. y 2 16, ( To find the roots of a quadratic equation, plug its coefficients (, and ) into the quadratic formula: 7 additional steps.

+ Standard Form: y=ax^2+bx+c y = ax2 +bx+ c 2. x ( In math, a quadratic equation is a second-order polynomial equation in a single variable. Ex 4.4 ,2Find the values of k for each of the following quadratic equations, so that they have two equal roots. a 25, ( 14 If a is 1 the coefficients may be read off directly. x v In his work Arithmetica, the Greek mathematician Diophantus solved the quadratic equation, but giving only one root, even when both roots were positive. 2 4 + < Medium Solution Verified by Toppr Px (x-3) + 9 = 0 px2-3px+9=0 a=p , b=-3p , c=9 b2-4ac= (-3p)2-4 (p) (9) =9p2-36p for having equal roots b2-4ac = 9p2-36p = 0 9p2-36p = 0 9p (p-4) = 0 9p=0 (or) p-4=0 p=0 (rejected) or p=4 + ) 2 5 8 121 m Complex roots occur in the solution based on equation [5] if the absolute value of sin 2p exceeds unity. and the minus sign if Since the graph is symmetric with respect to a vertical line through the vertex, the vertex's x-coordinate is located at the average of the roots (or intercepts). 4 When the Discriminant ( b24ac) ) ( = = +

1 3 }, The quadratic equation may be solved geometrically in a number of ways. 32, 4 = With a purely geometric approach Pythagoras and Euclid created a general procedure to find solutions of the quadratic equation. 7 + + If the parabola is tangent to the x-axis, there is a double root, which is the x-coordinate of the contact point between the graph and parabola. 2 2 0. Learn faster and smarter from top experts, Download to take your learnings offline and on the go. =

3 ( = + The Square Root Property states If x2=kx2=k, What will happen if k<0?k<0? 30, 5 WebFind the value of P so that the quadratic equation px (x-3)+9=0 has two equal roots. Euclid, the Greek mathematician, produced a more abstract geometrical method around 300 BC. = If the discriminant is positive there are two solutions, if negative there is no solution, if equlas 0 there is 1 solution. a 9 A circle is drawn with the start and end point SC as a diameter. 16 ) + 4 x 30 2 For equations with real solutions, you can use the graphing tool to visualize the solutions.

7 8 2 a 26, 2

This produces the reduced quadratic equation:[12]. Perform any multiplication or division, from left to right: = Quadratic equations differ from linear equations by including a quadratic term with the variable raised to the second power of the form ax2.

+ 2 In order to use the Square Root Property, the coefficient of the variable term must equal one. 2 ) Two Distinct Real Roots. m + ) 2 ( 11 Apply the Quadratic Formula x2-14x+49 = A Quadratic Equation has two roots, and they depend entirely upon the discriminant. ) , [16] The graph of any quadratic function has the same general shape, which is called a parabola. Les rcepteurs DAB+ : postes, tuners et autoradios Les oprateurs de radio, de mux et de diffusion. Rewrite the radical as a fraction of square roots. 9, 36 x 36 ( 4 2 When solving quadratic equations, the term \ ( {b^2} 4ac\) is used. b 46 Experts are tested by Chegg as specialists in their subject area. 4 64 8 WebThe mathematical representation of a Quadratic Equation is ax+bx+c = 0. Les metteurs TNT, leurs caractristiques et leurs zones de couverture, Rception de la TNT en maison individuelle, Rception de la TNT en collectif (immeubles, lotissements, htels), La TNT dans les tablissements recevant du public (htels, hpitaux), Les rcepteurs avec TNT intgre (crans plats), Les adaptateurs pour recevoir la TNT gratuite en SD ou HD, Les terminaux pour les offres de la TNT payante, Les autres chanes et services du satellite, cble, TV par Internet, Les offres incluant les chanes de la TNT, Le matriel (dcodeurs, paraboles, accessoires ), La technique et la technologie de la TV par satellite, La technique et la technologie de la TV par le cble, La rception TV par Internet et rseaux mobile (3G/4G/5G), L'actualit des offres TV par Internet et rseaux mobile, Les offres TV des rseaux mobile 3G/4G/5G, La technique et la technologie de la TV par ADSL et fibre, La technique et la technologie de la TV sur les rseaux mobile, Meta-Topic du forum de la radio Numrique, Les zones de couverture et la rception DAB+. c If one of = 30. + So, when {b^2-4ac} = 0, the equation has two equal roots, which is equal to 0, x 2 30 = We notice the left side of the equation is a perfect square trinomial. 2 2 See quadratic residue for more information about extracting square roots in finite fields. 7 [25], In 628 AD, Brahmagupta, an Indian mathematician, gave the first explicit (although still not completely general) solution of the quadratic equation ax2 + bx = c as follows: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value." Write the left side as a square and simplify the right side if necessary. + = It is written in the form: ax^2 + bx + c = 0 where x is the variable, and a, b, and c are constants, a 0.

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