Solving Quadratics by Factoring

6.1: Solving Quadratics by Factoring. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 83 parts Practice Flashcards

1/3

Zero-product solutions

Objective: Solve a quadratic already written as a product.

Prerequisite skills: factoring quadratics and equation balance. If a product of real factors is zero, at least one factor is zero. Set each factor equal to zero separately. A nonzero constant multiplier does not create another root. Common error: setting both factors equal to zero simultaneously. Restrictions and interpretation: The zero-product property applies to a product equal to zero; list all real solutions.

Formulas

$AB = 0 ⇒ A = 0 or B = 0$

Worked examples

Try itWhat is the greater solution of (x − 22)(x − 25) = 0?Show answer

Set x − 22 = 0 or x − 25 = 0. The solutions are 22 and 25; the greater is 25. Both make one original factor zero. Therefore the requested answer is 25.

Swipe sideways to move between sections

Zero-product solutions

Objective: Solve a quadratic already written as a product.

Formulas

$AB = 0 ⇒ A = 0 or B = 0$

Worked examples

Try itWhat is the greater solution of (x − 22)(x − 25) = 0?Show answer

Set x − 22 = 0 or x − 25 = 0. The solutions are 22 and 25; the greater is 25. Both make one original factor zero. Therefore the requested answer is 25.

Factor before solving

Objective: Move all terms to one side and factor.

The product-zero rule requires zero on one side. Rearrange the equation into standard form before factoring; then solve each factor. Factoring an expression alone is not yet a solution set. Common error: applying the zero-product rule when the product equals a nonzero number. Restrictions and interpretation: The zero-product property applies to a product equal to zero; list all real solutions.

Formulas

$x² − (a+b)x + ab = 0 ⇒ x = a or b$

Worked examples

Try itWhat is the smaller solution of x² − 51x = -648?Show answer

Add 648 to both sides: x² − 51x + 648 = 0. Factor: (x − 24)(x − 27) = 0. The smaller root is 24. Therefore the requested answer is 24.

Do not divide away a root

Objective: Preserve a possible zero solution during factoring.

When all terms share x, factor it rather than dividing by x. Division by x would assume x is nonzero and could discard a valid root. A repeated factor gives one distinct root even if it occurs twice. Common error: dividing by x before checking whether x = 0 works. Restrictions and interpretation: The zero-product property applies to a product equal to zero; list all real solutions. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$x² − kx = x(x − k)$

Worked examples

Try itWhat is the sum of all distinct real solutions of 3x² − 81x = 0?Show answer

Factor 3x[x − 27] = 0. The distinct solutions are 0 and 27; their sum is 27. The root zero must not be discarded. Therefore the requested answer is 27.

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Next lessonSquare Roots and Completing the Square