Zero Product Property & Roots

TipLearning Objectives

By the end of this lesson, you’ll be able to:

  • Use the zero product property to solve factored quadratic equations.
  • Connect solutions (roots) to \(x\)-intercepts on the graph.
  • Recognize repeated factors and understand multiplicity.

Key Ideas

The Zero Product Property says:

If \(AB = 0\), then \(A = 0\) or \(B = 0\) (or both).

For a factored quadratic: \[ a(x - r_1)(x - r_2) = 0, \] the solutions are: \[ x = r_1, \quad x = r_2. \]

These solutions correspond to:

  • roots
  • zeros
  • solutions
  • \(x\)-intercepts of the graph

If a factor repeats, like \((x - r)^2\), the root has multiplicity 2, and the graph touches the \(x\)-axis instead of crossing.

Root multiplicity affects how a graph behaves at an x-intercept. A root with multiplicity 1 crosses the x-axis, while a root with multiplicity 2 touches and turns around.

Common Problem Types

1. Solve Already-Factored Quadratics

Set each factor equal to zero.

2. Factor First, Then Solve

Useful when the quadratic is expanded.

3. Repeated Roots

When a factor appears more than once, the solution repeats (multiplicity).

4. Difference of Squares

Use \(A^2 - B^2 = (A - B)(A + B)\) before solving.

Strategies

  • Always check if the equation is already factored—if not, factor first.
  • Set each factor equal to zero and solve.
  • Look for special factoring patterns (GCF, difference of squares).
  • After solving, connect results to the graph: do the \(x\)-intercepts match?
  • Expand your factors mentally if unsure—quick check for correctness.

Worked Examples

Example 1 — Solve a Factored Equation

Solve: \[ (x - 3)(x + 5) = 0 \]

Solution:

Set each factor equal to zero: - \(x - 3 = 0 \Rightarrow x = 3\)
- \(x + 5 = 0 \Rightarrow x = -5\)

Solutions: \(x = 3,\,-5\)

Example 2 — Factor Then Solve

Solve: \[ x^2 - x - 6 = 0 \]

Solution:

  1. Factor: numbers that multiply to \(-6\) and add to \(-1\) → \(-3\) and \(2\).
    \[ (x - 3)(x + 2) = 0 \]
  2. Set each equal to 0:
    • \(x = 3\)
    • \(x = -2\)

Solutions: \(x = 3,\,-2\)

Example 3 — Repeated Root (Multiplicity)

Solve: \[ (x + 4)^2 = 0 \]

Solution:

\((x + 4)^2 = (x + 4)(x + 4)\)

Set one factor equal to zero: \[ x + 4 = 0 \Rightarrow x = -4 \]

This root has multiplicity 2.
Graphically: the parabola touches the \(x\)-axis at \(x = -4\).

WarningCommon Mistakes
  • Dividing by a factor instead of setting it equal to zero.
  • Forgetting one factor, leading to missing a solution.
  • Not factoring completely before applying the zero product property.

Practice Problems

  1. Solve: \((x + 1)(x - 4) = 0\)
  2. Solve: \(x^2 + 5x + 6 = 0\)
  3. Solve: \(2x^2 - 8x = 0\)
  4. Solve: \((3x - 2)^2 = 0\)
  5. Solve: \(x^2 - 9 = 0\)

1.
\(x + 1 = 0 \Rightarrow x = -1\)
\(x - 4 = 0 \Rightarrow x = 4\)
Solutions: \(x = -1,\,4\)

2.
Factor: \((x + 2)(x + 3)\)
Solutions: \(x = -2,\,-3\)

3.
Factor GCF: \(2x(x - 4) = 0\)
Solutions: \(x = 0,\,4\)

4.
\((3x - 2)^2 = 0 \Rightarrow 3x - 2 = 0\)
\(x = \frac{2}{3}\) (multiplicity 2)

5.
Difference of squares: \((x - 3)(x + 3) = 0\)
Solutions: \(x = 3,\,-3\)

Summary

  • Use the zero product property by setting each factor equal to zero.
  • Factoring is the essential first step unless the equation is already factored.
  • Each factor gives a solution; repeated factors create repeated roots.
  • Roots correspond directly to the graph’s \(x\)-intercepts.
  • If it’s factored, set each factor equal to zero—done.
  • If it’s not factored, factor completely before solving.
  • Look for patterns: GCF, trinomials, difference of squares.
  • Repeated factors mean the graph touches (not crosses) the axis.