__(A) Identifying quadratic expression__**1**. A quadratic expression is an algebraic expression of the form

*ax*

^{2}+

*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, 5

*x*

^{2}– 6

*x*+ 3 is a quadratic expression.

**Example 1**

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

(a)

*x*^{2}– 5*x*+ 3
(b) 8

*p*^{2}+ 10
(c) 5

*x*+ 6
(d) 2

*x*^{2}+ 4*y*+ 14
(e) $2p+\frac{1}{p}+6$

(f)

*y*^{3}– 3*y*+ 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.

**(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.

**(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.

(2

*x*+ 3)(*x*- 3) = 2*x*^{2}– 3*x*– 9**Example 2**

Multiply the following pairs of linear expressions.

(a) (4

*x*+ 3)(*x*– 2)
(b) (

*y*– 6)^{2}
(c) 2

*x*(*x*– 5)

**Solution:****(a)**(4

*x*+ 3)(

*x*– 2)

= (4

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

*x*^{2}– 8*x*+ 3*x*– 6
= 4

*x*^{2}– 5*x*– 6**(b)**(

*y*– 6)

^{2}

= (

*y*– 6)(*y*– 6)
= (

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

*y*^{2}-6*y*– 6*y*+ 36
=

*y*^{2}- 12*y*+ 36**(c)**2

*x*(

*x*– 5)

= 2

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

*x*^{2}– 10*x*