2 Quadratic Equations 1

2. QUADRATIC EQUATIONSIMPORTANT NOTES :(i) The general form of a quadratic equation is ax2 + bx + c = 0; a, b, c are constants and a ≠ 0.(ii) Characteristics of a quadratic equation:

(a) Involves only ONE variable,(b) Has an equal sign “ = ” and can be expressed in the form ax2 + bx + c = 0,(c) The highest power of the variable is 2.

2.1 Recognising Quadratic Equations

EXAMPLESNo Quafratic Equations (Q.E.) NON Q.E. WHY?

1. x2 + 2x -3 = 0 2x – 3 = 0 No terms in x2 ( a = 0)

2. x2 = ½ x 2 2

= 0x

Term 2

x3. 4x = 3x2 x3 – 2 x2 = 0 Term x3

4. 3x (x – 1) = 2 x2 – 3x -1 + 2 = 0 Term x -1

5. p – 4x + 5x2 = 0, p constant x2 – 2xy + y2 = 0 Two variables

Exercise : State whether the following are quadratic Equations. Give your reason for Non Q.E.

No Function Q.F. Non Q.F. WHY?

0. 3x - 2 = 10 – x √ No terms in x2

1. x2 = 102

2. 12 – 3x2 = 0

3. x2 + x = 6

4. 2x2 + ½ x - 3 = 0

5. 6 = x

x

6. 0 = x ( x – 2)

7. 2x2 + kx -3 = 0, k constant

8. (m-1) x2 + 5x = 2m , m constant

9. 3 – (p+1) x2 = 0 , p constant

10. p(x) = x2 + 2hx + k+1, h, k constants

11. f(x) = x2 – 4

12. (k-1)x2 – 3kx + 10 = 0 , k constant

2 Quadratic Equations 2

2.2 The ROOTs of a quadratic Equation (Q.E.)

Note : “ROOT” refers to a specific value which satisfies the Q.E.

Example : Given Q.E. x2 + 2x – 3 = 0

By substitution, it is found that :x = 1 , 12 + 2(1) – 3 = 0

Hence 1 is a root of the quadratic equation x2 + 2x – 3 = 0.But if x = 2, 22 + 2(2) – 3 ≠ 0,

We say that 2 is NOT a root of the given quadratic Equation.

EXAMPLE EXERCISEC1. Determine if -2 is a root of the equation

3x2 + 2x -7 = 0.

x = -2, 3(-2)2 + 2(-2) – 7 = 12 – 4 – 7≠ 0

Hence - 2 is NOT a root of the given equation.

L1. Determine if 3 is a root of the equation 2x2 – x – 15 = 0.

L2. L1. Determine if 3 is a root of the equation x2 – 2x + 3 = 0.

L3. Determine if ½ is a root of the equation 4x2 + 2x – 2 = 0.

C2. If -2 is a root of the quadratic equation x2– kx – 10 = 0, find k.

x = -2, (-2)2 – k(-2) – 10 = 0-4 + 2k – 10 = 0

2k = 14k = 7

L4. If 3 is a root of the equation x2– 2kx + 12 = 0 , find k.

L5. If -2 and p are roots of the quadratic equation 2x2 + 3x + k = 0, find the value of k and p.

L6. If -1 are roots of the quadratic equation px2 – 4x + 3p – 8 = 0, find p.