Quadratic EquationMind MapQuadratic Equation

Identify QE

Solving QE

Forming QE from Roots

Type of Root

Example of QE and non-QE

General Form

Root of QE

3 Methods

x2 SoRx + PoR = 0

Value of a, b and c

Substitution

Factorisation

SoR = -b/a PoR = c/a

Intersection of Graph

Completing the Square

A few example of SoR and PoR

Formula

Quadratic Equation In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

Difference Between Quadratic Eqaution and Quadratic Function

ax 0

2

+ bx + c =

where a 0. (For a = 0, the equation becomes a linear equation.) The letters a, b, and c are called coefficients: the quadratic coefficient a is the coefficient of x2, the linear coefficient b is the coefficient of x, and c is the constant coefficient, also called the free term or constant term.

Example of Quadratic Equation 2 x2 5 = 0 1 6 x2 = 3 6 x + 3x 2 = 0 x2 = 0

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Quadratic EquationSolving Quadratic Equation 3 Methods: Factorisation Completing The Square Quadratic Formula Factorisation

Example Solve x2 + 5x + 6 = 0. Answer x2 + 5x + 6 = (x + 2)(x + 3) Set this equal to zero: (x + 2)(x + 3) = 0 Solve each factor: x + 2 = 0 or x + 3 = 0 x = 2 or x = 3 The solution of x2 + 5x + 6 = 0 is x = 3, 2

Completing The Square

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Quadratic EquationQuadratic Formula

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Quadratic EquationForming Quadratic Equation from Its Roots

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Quadratic Equation

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Quadratic Equation

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Quadratic Equation

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Quadratic Equation

Nature of Roots of a Quadratic Equation 2The expression b 4ac in the general formula is called the discriminant of the equation, as it determines the type of roots that the equation has. b 2 4ac> 0 two real and distinct roots

b 2 4ac= 0 two real and equal roots b 2 4ac b 2 4ac< 0 no real roots 0 the roots are real

e.g. 1: e.g. 3: Find the range of values of k for which the equationFind the range of values of p for which the equation 2x 2 + 5x + 3 k = 0 has two real distinct roots.x 2 2 px + p 2 + 5 p 6 = 0 has no real roots. b 2 4ac > 0 (5) 2 4(2)(3 k ) > 0 25 24 + 8k > 0 1 + 8k > 0 8k > 1 1 k> 8 e.g. 2: b 2 4ac < 0 (2 p) 2 4(1)( p 2 + 5 p 6) < 0 4 p 2 4 p 2 20 p + 24 < 0 20 p + 24 < 0 20 p < 24 20 p > 24 24 p> 20 6 p> 5

2The roots of 3x + kx + 12 = 0 are equal. Find k. b 2 4ac = 0 (k ) 2 4(3)(12) = 0

k 2 144 = 0 k 2 = 144 k = 144 k = 12

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Quadratic Equatione.g. 4:2 2Show

+2=0always has real roots. b 2 4ac = (3a ) 2 4(a 2 )(2) = 9 a 2 8a = a22

that the equation a x + 3ax

a2 >0 for all values of a. Therefore b 2 4ac > 0 Proven that

a 2 x 2 + 3ax + 2 = 0 always has real roots.

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