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Solving Quadratic Equations by Factoring

Equations that can be written in the form Ax2 + Bx + C = 0

where a, b, and c are integers and a>0 are called quadratic equations . This form is called
the standard form.

The easiest way to solve quadratic equations is to factor (if possible) the polynomial on the left
side of the equation above. We then use the Zero- Product Theorem .

Zero-Product Theorem for any real numbers
a and b, if ab = 0, then a = 0 or b = 0.

Quadratic Equations:

1. x2 – 3x – 4 = 0 This quadratic equation is in standard form.
2. A2 = 6A – 9 This quadratic equation is not in standard form.
3. (x – 7)(x + 6) = –22 This is a quadratic equation not in standard form.
4. (x + 8)(x – 5) = 0 By the Zero-Product Theorem, the quadratic equation can be
solved.
5. x(x – 14) = 0 This quadratic equation is in factored form and equal to zero ;
it can be solved by using the Zero-Product Theorem.
To solve quadratic equations by factoring:

1. Put the equation into standard form.
2. Factor the polynomial.
3. Set each factor equal to zero.
4. Solve each of the first degree equations.
EXAMPLE 1: 2x2 – 9x – 35 =0
Solution: This is already in standard form, so we start by factoring.
Factor

Set each factor = 0
Solve each equation
The solution set is:
EXAMPLE 2: a2 = 6a – 9
Solution: We first put this into standard form.
The equation is not in
standard form
a2 = 6a – 9
Put into standard form a2 – 6a + 9 = 0
Factor
Set each factor = 0
Solve each equation
The solution set is: {3,3}
EXAMPLE 3: (t – 7)(t + 6) = –22
Solution: Multiple this out and put into standard form.
FOIL
Standard form
Factor
Set each factor = 0
Solve each equation t = 5 t = –4
The solution set is: {5,–4}
EXAMPLE 4: Solve x(x – 14) = 0
Solution The equation is in factored form and equal to zero. It can be
solved by using the Zero-Product Theorem.
Factor
Set each factor = 0
The solution set is: {0,14}
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