GED Mathematical Reasoning
Quadratics & Factoring
A quadratic has a squared variable in it, like x² + 5x + 6. Because of that square, it usually has two answers, not one. Factoring means rewriting it as two parentheses multiplied together, which is what lets you find those answers.
The words first
- quadratic
- an expression whose highest power is 2
- factor
- to rewrite as a multiplication
- roots / solutions / zeros
- three names for the same thing — the x values that make it equal 0
- FOIL
- First, Outer, Inner, Last — how you multiply two parentheses back out
The method
FACTORING x² + bx + c
Find two numbers that MULTIPLY to c and ADD to b.
x² + 5x + 6
multiply to 6, add to 5 → 2 and 3
(x + 2)(x + 3)
SIGNS TELL YOU WHAT TO LOOK FOR
+ c and + b → both numbers positive: (x+2)(x+3)
+ c and − b → both negative: (x−2)(x−3)
− c → one of each: (x+4)(x−2)
SOLVING ONCE IT IS FACTORED
If two things multiply to zero, one of them must BE zero.
(x + 2)(x + 3) = 0
x + 2 = 0 → x = −2
x + 3 = 0 → x = −3
CHECKING BY FOIL
(x+2)(x+3) = x² + 3x + 2x + 6 = x² + 5x + 6 ✓
Worked all the way through
The problem
Solve x² − 7x + 12 = 0.
- Step 1 — Read the signs first.The last number is positive (+12) and the middle is negative (−7). Positive last with negative middle means both numbers are negative.
- Step 2 — Find two numbers that multiply to 12.Options: 1 and 12, 2 and 6, 3 and 4.
- Step 3 — Which pair adds to 7?3 + 4 = 7. That is the pair.
- Step 4 — Write them as negatives, since step 1 said both negative.(x − 3)(x − 4) = 0
- Step 5 — Set each parenthesis to zero.x − 3 = 0 → x = 3 x − 4 = 0 → x = 4
- Step 6 — Check one.3² − 7(3) + 12 = 9 − 21 + 12 = 0 ✓
On the board
x² − 7x + 12 = 0
the problem
? × ? = 12 and ? + ? = −7
what pair fits both?
−3 and −4
−3 × −4 = 12, −3 + −4 = −7
(x − 3)(x − 4) = 0
write the factors
x − 3 = 0 → x = 3
each factor can be the zero
x − 4 = 0 → x = 4
Answer
x = 3 and x = 4
A second one, in a different form
Solve x² + 5x + 6 = 0.
x² + 5x + 6 = 0
the problem
? × ? = 6 and ? + ? = 5
+6 and +5 → both positive
2 and 3
2 × 3 = 6, 2 + 3 = 5
(x + 2)(x + 3) = 0
x = −2 or x = −3
each factor set to zero
Answer
x = −2 and x = −3
The usual mistake
Getting the signs backward when factoring.
Check yourself
Notice the answers came out positive even though the parentheses had minus signs. Setting x − 3 = 0 gives x = +3. Reading the sign straight off the parenthesis is the usual mistake.
Practice it
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