Mathematics 9 Lesson 1-C: Roots and Coefficients of Quadratic Equations

ROOTS AND COEFFICIENTS OF

QUADRATIC EQUATIONS

DISCRIMINANTThe radicand () in the quadratic formula

USING THE DISCRIMINANTGiven a quadratic equation in the form of ax + bx + c = 0, where a, b, and c are real numbers and a≠0. We can determine the number and type of solutions of a quadratic equation, by evaluating the discriminant

USING THE DISCRIMINANT1. If > 0, the equation has two real

solutions. Both will be rational if the discriminant is a perfect square or irrational, otherwise

USING THE DISCRIMINANT2. If = 0, the equation has only

one solution which will be a rational number

USING THE DISCRIMINANT3. If < 0, the equation has no real

number solution

USING THE DISCRIMINANTExample: 16x - 8x + 1 = 0 = (-8) – 4(16)(1) = 64 – 64 = 0Because the discriminant is 0, the equation has only solution

RELATION OF ROOTS1.The sum of the roots is the

additive inverse of the quotient of b and a

r1 + r2 = -

RELATION OF ROOTS2.The product of the roots is the

quotient of c and ar1 - r2 =

RELATION OF ROOTSThe relations that exist between the roots of a quadratic equation which can be used in checking the validity of the roots can be of best use in deriving the quadratic equation

General Form: ax2 + bx + c = 0Expressed in the form of Multiplication Property of Equality:x2 + + = 0

RULETo derive the quadratic equation when the two roots are given, subtract each root from x to get the corresponding linear factors and equate the product of the linear factors to zero

Example: x2 – 3x – 28 = 0Solution:(x-7) (x+4) = 0x-7 = 0 or x+4=0x=7 (root 1) x=-4 (root 2)

Example: x2 – 3x – 28 = 0Check whether the roots are correct:r1 + r2 = - and r1 - r2 = (7) + (-4) = (7) (-4) = - 3 = 3 True -28 = -28 True