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Quadratic Equations

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Quadratic Equation is a polynomial equation of second degree. The general form of a quadratic equations is ax2+bx+c = 0. The contributions of the ancient Indian Mathematicians to quadratic equations are quite significant and extensive. Before 800BC Indian Mathematicians constructed 'altars' based on the solutions of quadratic equation ax2+bx+c =0, Aryabhatta gave a rule to sum the geometric series which involves the solution of a quadratic equation. Discriminant of Quadratic Equation The following table shows the nature of the roots of a quadratic equation with rational coefficients. Discriminant Square root = 0 perfect rational and equal > 0 rational and unequal perfect or not (or) Perfect irrational and unequal roots not perfect complex and conjugate roots in pair

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QuadraticEquations
QuadraticEquationisapolynomialequationofseconddegree.Thegeneralformofaquadraticequationsis
ax2+bx+c = 0. The contributions of the ancient Indian Mathematicians to quadratic equations are quite
significantandextensive.
Before 800BC Indian Mathematicians constructed 'altars' based on the solutions of quadratic equation
ax2+bx+c=0,Aryabhattagavearuletosumthegeometricserieswhichinvolvesthesolutionofaquadratic
equation.
DiscriminantofQuadraticEquation
Thefol owingtableshowsthenatureoftherootsofaquadraticequationwithrationalcoefficients.
Discriminant
Square
root
=0
perfect
rationalandequal
>0
rationalandunequal
perfectornot
(or)
Perfect
irrationalandunequalroots
<0
notperfect
comple
xandconjugaterootsinpair
LearnMoreaboutSolvingQuadraticEquations

FormationofaQuadraticEquation
Letusformthequadraticequationwhoserootsare[alpha]and[beta].
Thenx=[alpha],y=[beta]aretheroots
Thereforex[alpha]=0and,y[beta=0]
so(x[alpha])(y[beta][)=0]
MaximumandMinimumValuesofaQuadraticExpression
Anexpressionofthetypeax2+bx+ciscalled"quadraticexpression".
Thequadraticexpressionax2+bx+ctakesdifferentvaluesasxtakesdifferentvalues.
SolvingQuadraticEquations
HereistheexamplesonSolvingaquadraticequationbasedonmethodstosolveit
FactoringMethod
Example:1Solvex2+2x=15byfactoring.

ReadMoreonHowToSolveQuadraticEquations

Asxvariesfrom[prop]to+[prop]ax2+bx+c
1.hasaminimumvaluewhenevera>0.
Theminimumvalueofthequadraticexpressionis(4acb2)/4aanditoccursatx=b/2a.
2.hasamaximumvaluewhenevera<0.
Themaximumvalueofthequadraticexpressionis(4acb2)/4aanditoccursatx=b/2a.
QuadraticEquationFormula
Thegeneralformofaquadraticequationsisax2+bx+c=0.
Thesetofallsolutionsofaquadraticequationiscalleditssolutionset.Thevaluesofxthatmake
aquadraticequationtrueiscalleditsrootsorzerosorsolutions.Quadraticequationscanbe
solvedbyfactorizationmethodorbyusingquadraticformula
x=(b(b4ac))/(2a)
quadraticformula
[x=(b+sqrt(b]2[4ac))/(2a)]
[wh ereb]2[4ac][iscalledthediscriminantof
thequadraticequation.][Aquadraticequation
hastworoots.]

Rewriteequationinstandardform:x2+2x15=0
Factortheleftside:(x+5)(x3)=0
Applyzeroproductrule:x+5=0orx3=0
Solveforxineachequation:x=5orx=3
SquareRootMethod
Example2:Solveequation(3x1)29=0.
Applysquarerootmethod:(3x1)2=9
3x1=[sqrt9]or3x1=[sqrt9]
3x1=3or3x1=3
Solveequations:
3x1+1=3+1or3x1+1=3+1
3x=4or3x=2
X=[4/3]orx=[2/3]

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