Solving by Factoring

Solve quadratic equations by factoring method. Math 20-1 Alberta Grade 11 curriculum.

Lesson 1.7 of Quadratic Functions & Equations in Math 20-1 — Alberta curriculum lessons.

What you'll learn in this lesson

• The Method — Why Factoring Works. Discover how x-intercepts of a parabola are exactly the solutions to the corresponding quadratic equation, then learn the step-by-step process for solving by factoring.
• Basic Factoring Examples. Practice applying the Zero Product Property to already-factored expressions and to simple trinomials and difference-of-squares equations.
• Trickier Equations. Handle equations with fractions, rearrangement, hidden cubics, and a substitution trick for disguised quadratics.
• Building Equations and Sketching from Roots. Reverse the process to write a quadratic equation from given roots, then use intercepts to sketch the parabola.

Inside the lesson — a free preview

The Connection: Function vs. Equation

Two Ways of Asking the Same Question. A quadratic function has two variables (x and y) and can be graphed.

A quadratic equation has one variable and can be solved.

The x-intercepts of the function y = ax^2 + bx + c are the same numbers as the solutions to the equation 0 = ax^2 + bx + c. They are just two ways of asking the same question.

Worked example: See the Connection — y = 3x^2 - 5x - 2

Part a) — Find the x-intercepts graphically. Graph the function y = 3x^2 - 5x - 2 and read the x-intercepts from the graph.

Part b) — Write the corresponding quadratic equation. To find x-intercepts, set y = 0:

0 = 3x^2 - 5x - 2

Part c) — Solve algebraically by factoring. Step 1 — Factor 3x^2 - 5x - 2 using decomposition.

a × c = (3)(-2) = -6

Need two numbers multiplying to -6 and adding to -5 → -6 and 1.

Split the middle term: 3x^2 - 6x + x - 2

Group: 3x(x - 2) + 1(x - 2) = (x - 2)(3x + 1)

So: 0 = (x - 2)(3x + 1)

Step 2 — Apply the Zero Product Property.

If two factors multiply to 0, at least one of them must equal 0.

x - 2 = 0 → x = 2

3x + 1 = 0 → 3x = -1 → x = -(1)/(3)

Solutions: x = 2 and x = -(1)/(3)

Part d) — What do you notice?. The graphical x-intercepts (x ≈ -0.33 and x = 2) match the algebraic solutions (x = -(1)/(3) and x = 2) exactly.

Key insight: Finding x-intercepts of a function and solving the corresponding quadratic equation are the same thing.

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