Factoring and Quadratics Basics
Key rules, patterns, and steps for factoring polynomials and solving quadratic equations.
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Questions Covered in This Set
12 cards to master
Zero Product Property
If A · B = 0, then A = 0 or B = 0. This is why factoring solves equations — but the equation must equal zero first.
What is the first step in ANY factoring problem?
Pull out the GCF (greatest common factor) from every term. Example: 4x³ − 12x² + 8x = 4x(x² − 3x + 2) = 4x(x − 1)(x − 2).
How do you factor x² + bx + c?
Find two numbers that multiply to c and add to b. Example: x² + 7x + 12 → 3 and 4 → (x + 3)(x + 4).
Sign shortcut for trinomials
If c is positive, both numbers share the sign of b. If c is negative, the numbers have opposite signs and the larger one carries the sign of b.
Difference of squares pattern
a² − b² = (a + b)(a − b). Example: 9x² − 25 = (3x + 5)(3x − 5). Note: x² + 49 does NOT factor over the reals.
Perfect square trinomial patterns
a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². Example: x² + 10x + 25 = (x + 5)².
Three steps to solve a quadratic by factoring
1) Move everything to one side so it equals 0. 2) Factor. 3) Set each factor equal to 0 and solve.
Why should you never divide both sides by a variable?
You lose solutions. 3x² = 12x → dividing by x loses x = 0; factoring gives 3x(x − 4) = 0 → x = 0 or x = 4.
What do the solutions mean graphically?
They are the x-intercepts (roots/zeros) of the parabola y = ax² + bx + c — where the U-shaped curve crosses the x-axis.
How many real solutions can a quadratic have?
Two (crosses twice), one repeated (touches at vertex, e.g. x² − 6x + 9 = 0 → x = 3), or none (e.g. x² + 4 = 0).
Solve x² + 5x = 14
x² + 5x − 14 = 0 → (x + 7)(x − 2) = 0 → x = −7 or x = 2.
Quadratic formula
x = (−b ± √(b² − 4ac)) / (2a). Always works, even when a quadratic won't factor with nice integers.
