➕ Algebra 1 · Quadratics
Solving Quadratics by Factoring flashcards
Say the answer out loud before you flip. Cards you miss come back sooner; cards you know come back later. That spacing is what builds long-term memory.
All 17 flashcards at a glance
Card 1
What is a quadratic equation?
An equation that can be written as ax2 + bx + c = 0, where a ≠ 0.
Card 2
State the zero-product property.
If A × B = 0, then A = 0 or B = 0 (or both).
Card 3
Why must you get 0 on one side before using the zero-product property?
The property works only when the product equals 0. If a product equals 12 (or any other number), neither factor has to equal that number.
Card 4
What are the five steps for solving a quadratic equation by factoring?
Zero on one side, factor completely, set each factor to 0, solve each one, and check in the original equation.
Card 5
Solve (x − 2)(x + 9) = 0.
x − 2 = 0 or x + 9 = 0, so x = 2 or x = −9.
Card 6
Solve x2 − 5x + 4 = 0.
Factor: (x − 1)(x − 4) = 0. So x = 1 or x = 4.
Card 7
Solve x2 − 3x = 0.
Factor out x: x(x − 3) = 0. So x = 0 or x = 3.
Card 8
A student solves x2 = 4x by dividing both sides by x and gets x = 4. What went wrong?
Dividing by x is only allowed when x ≠ 0, and it loses the solution x = 0. Correct: x2 − 4x = 0, so x(x − 4) = 0, so x = 0 or x = 4.
Card 9
What is a double root? Give an example.
A solution that comes from a factor appearing twice. x2 − 6x + 9 = (x − 3)2 = 0 has the single solution x = 3.
Card 10
In 7x(x − 3) = 0, why does the factor 7 give no solution?
Because 7 can never equal 0. Only the factors that can be 0, x and x − 3, give equations: x = 0 or x = 3.
Card 11
How are the solutions of ax2 + bx + c = 0 related to the graph of y = ax2 + bx + c?
They are the x-intercepts: the x-values where the graph meets the x-axis, at the points (p, 0) where y = 0.
Card 12
Can you solve x2 + 4x + 1 = 0 by factoring with integers?
No. No two integers multiply to 1 and add to 4 (1 and 1 add to 2, and −1 and −1 add to −2), so the trinomial is prime. The equation still has solutions, which later lessons find.
Card 13
In an area problem you get w = −5 or w = 9 for a width. Which do you keep, and why?
Keep w = 9. A width is a length, so it cannot be negative. Reject w = −5 and say why.
Card 14
Solve x2 − 36 = 0.
Difference of squares: (x − 6)(x + 6) = 0. So x = 6 or x = −6.
Card 15
Solve −x2 + 9x − 14 = 0.
Multiply both sides by −1 (not 0): x2 − 9x + 14 = 0. Factor: (x − 2)(x − 7) = 0. So x = 2 or x = 7.
Card 16
Why do you check a solution in the original equation, not the factored form?
Checking the original equation also catches mistakes made while moving terms or factoring. A check in the factored form cannot reveal an error in the factoring itself.
Card 17
In this lesson’s “Zebras Find Safe Shady Corners” trick, what do the first letters stand for?
Zero on one side, Factor completely, Set each factor to 0, Solve each one, Check in the original equation.