1.1 Quadratic Expressions - SPM Mathematics

(A) Identifying quadratic expression

1. A quadratic expression is an algebraic expression of the form ax² + bx + c, where a, b and c are constants, a ≠ 0 and x is an unknown.

(a) The highest power of x is 2.

(b) For example, 5x²– 6x + 3 is a quadratic expression.

Example 1

State whether each of the following is a quadratic expression in one unknown.

(a) x² – 5x + 3

(b) 8p² + 10

(d) 2x² + 4y + 14

(e)

2 p + \frac{1}{p} + 6

(f) y³ – 3y + 1

Solution:

(a) Yes. A quadratic expression in one unknown.

(b) Yes. A quadratic expression in one unknown.

(c) Not a quadratic expression in one unknown. The highest power of the unknown x is not 2.

(d) Not a quadratic expression in one unknown. There are 2 unknowns, x and y in the quadratic expression.

(e) Not a quadratic expression in one unknown. The highest power of the unknown x is not 2.

\frac{1}{p} = p^{- 1}

(f) Not a quadratic expression in one unknown. The highest power of the unknown x is not 2.

2. A quadratic expression can be formed by multiplying two linear expressions.

(2x + 3)(x – 3) = 2x² – 3x – 9

Example 2

Multiply the following pairs of linear expressions.

(a) (4x + 3)(x – 2)

(b) (y – 6)²

Solution:

(a) (4x + 3)(x – 2)

= (4x)(x) + (4x)(-2) +(3)(x) + (3)(-2)

= 4x² – 8x + 3x – 6

= 4x² – 5x – 6

(b) (y – 6)²

= (y – 6)(y – 6)

= (y)(y) + (y)(-6) + (-6)(y) + (-6)(-6)

= y² -6y – 6y + 36

= y² – 12y + 36

(c) 2x (x – 5)

= 2x(x) + 2x(-5)

= 2x² – 10x