5.3: Solving Quadratic Equations
(Factoring)
• Factoring is a way to break up a quadratic
equation into different pieces.
• Ex. 12 can be “factored” as 3 x 4, or 2 x 6
• Quadratics can usually be factored as the
product of two terms.
Factoring - GCF
• Greatest Common Factor (GCF): 6𝑥 2 − 15𝑥
1. Determine the greatest number factor of
each term.
2. Determine the greatest exponent of 𝑥 that
each term shares.
3. Write these on the outside:
Factoring - GCF
4. Divide each term by the GCF and write what
is “leftover” in the parentheses.
Factoring - GCF
• Factor each of the following by using the GCF
1. 3𝑥 3 + 6𝑥 2 − 18𝑥
2. −𝑥 2 + 5𝑥 − 4
Difference of Perfect Squares
• When a quadratic is in the form 𝑎2 − 𝑏2 there
is a special way to factor.
• Difference of Perfect Squares Rule:
𝑎2 − 𝑏2 = (𝑎 + 𝑏)(𝑎 − 𝑏)
• Ex. 9𝑥 2 − 4 = (3𝑥 + 2)(3𝑥 − 2)
Difference of Perfect Squares
• Factor the following using difference of
perfect squares.
1. 𝑥 2 − 16
2. 4𝑥 2 − 1
Trinomials
• Quadratics in the form 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 are
called trinomials since they have three terms.
• To factor a trinomial, you are stuck guessing
and checking for a “reverse FOIL.”
Trinomials
• Ex. 𝑥 2 + 7𝑥 + 12
1. Find factors of c that will add/subtract to b.
2. Place the two numbers in the parentheses:
(𝑥
)(𝑥
)
Trinomials
• 𝑥 2 − 6𝑥 + 8
𝑥 2 − 3𝑥 − 15
Trinomials
• When 𝑎 is a number other than 1, factoring is
not as simple as before.
• Ex. 2𝑥 2 − 9𝑥 + 7
• Use the “Garbage Method” OR Guess and
Check
Garbage Method
2
Ex. 2𝑥 − 9𝑥 + 7
1. Multiply 𝑎 × 𝑐.
2. Find two factors of a x c that add/subtract to
your answer.
3. Write those factors into the form:
( 𝑥
)( 𝑥
)
4. Put 𝑎 in the front of each parentheses.
( 𝑥
)( 𝑥
)
Garbage Method
5. Take out the “garbage” (any common factors)
and write what is leftover.
(2𝑥 − 7)(2𝑥 − 2)
Garbage Method
• Factor:
6𝑥 2 − 7𝑥 − 3
Factoring Completely
• To factor completely, begin with GCF (if
necessary) and the continue to factor each
piece using other techniques: diff. of perfect
squares, trinomials, garbage.
Factoring Completely
• Factor completely:
1. 8𝑥 2 − 50
2. 2𝑥 3 − 2𝑥 2 − 24
Factoring Completely
• Factor: 10𝑥 4 − 45𝑥 3 + 25𝑥 2