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
(c) 5x + 6
(d) 2x2 + 4y + 14
(e)  $2p+\frac{1}{p}+6$
(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
(c) 2x (x – 5)

Solution:
(a) (4x + 3)(– 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

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