Chapter 10 QuadratiC EQuations 355

13. x

xx

xx

2 1

9

0

1

3

1

3

0

1

3

0 1

3

0

1

3

−=

 −







 +







=

−= +=

++

(^11)
3
1
3
1
3
1
3
1
3
−−
xx==−
Not all quadratic expressions are as easy to factor as the previous examples
and problems. Sometimes we have to multiply or divide both sides of the equa-
tion by some number in order to factor the nonzero side of the equation.
Because zero multiplied or divided by any nonzero number is still zero, only
one side of the equation changes. Keep in mind that not all quadratic expres-
sions can be factored using rational numbers (fractions) or even real numbers.
Later, we will learn another method of solving every quadratic equation which
bypasses the factoring method.
EXAMPLES
Solve the equation by factoring.
–x^2 + 4 x − 3 = 0
The equation −x^2 + 4 x − 3 = 0 is awkward to factor because of the negative
sign in front of x^2. Multiplying both sides of the equation by −1 makes the
factoring a little easier.
−− +−=−
−+=
−−=
14310
430
310
2
2
()()
()()
xx
xx
xx
xx
xx
−= −=
++ ++

30 10
33 11
31      
Decimals and fractions in a quadratic equation can be eliminated in the
same way; multiplying both sides of the equation either by a power of
EXAMPLES
Solve the equation by factoring.

EXAMPLES

Solve the equation by factoring.