1. Factoring quadratic expressions

2. Quadratic expressions
A quadratic expression is an expression in which the highest power of the variable is 2. For example, t2 2
x2 – 2,
w2 + 3w + 1,
4 – 5g2 ,
The general form of a quadratic expression in x is:
ax2 + bx + c (where a = 0)
x is a variable.
As well as the highest power being two, no power in a quadratic expression can be negative or fractional.
Compare each of the quadratic expressions given with the general form.
In x2 – 2, a = 1, b = 0 and c = –2.
In w2 + 3w + 1, a = 1, b = 3 and c = 1. This is a quadratic in w.
In 4 – 5g2, a = –5, b = 0 and c = 4. This is a quadratic in g.
In t2/2, a = ½, b = 0 and c = 0. This is a quadratic in t.
a is a fixed number and is the coefficient of x2.
b is a fixed number and is the coefficient of x.
c is a fixed number and is the constant term.

3. Factoring expressions
Remember: factoring an expression is the opposite of multiplying it.
Multiplying
Factoring
(a + 1)(a + 2)
a2 + 3a + 2
Often:
When we multiply an expression we remove the parentheses.
When we factor an expression we write it with parentheses.

4. Factoring quadratic expressions
Quadratic expressions of the form x2 + bx + c can be factored if they can be written using parentheses as
(x + d)(x + e)
where d and e are integers.
If we multiply (x + d)(x + e) we have:
(x + d)(x + e) = x2 + dx + ex + de
= x2 + (d + e)x + de
Students will require lots of practice to factor quadratics effectively.
This slide explains why when we factor an expression in the form x2 + bx + c to the form (x + d)(x + e) the values of d and e must be chosen so that d + e = b and de = c.
(x + d)(x + e) = x2 + (d + e)x + de is an identity. This means that the coefficients and constant on the left-hand side are equal to the coefficients and constant on the right-hand side.
Comparing this to x2 + bx + c we can see that:
The sum of d and e must be equal to b, the coefficient of x.
The product of d and e must be equal to c, the constant term.

5. Factoring quadratic expressions 1
Factor the given expression by finding two integers that add together to give the coefficient of x and multiply together to give the constant.
It may be a good idea to practice adding and multiplying negative numbers before attempting this activity.
The lesser of the two hidden integers will be given first in each case.

6. Matching quadratic expressions 1
Select a quadratic expression and ask a volunteer to find its corresponding factored form.