Notes on Zero Product Property and Solving Quadratic Equations

Zero Product Property and Quadratic Equations

Introduction

  • Multiplying any number by zero results in a product of zero.
  • This property can be used to solve quadratic equations.
  • Goal: Apply the zero product property to solve quadratic equations by factoring and writing quadratic equations given real number zeros.

Key Terms

  • Zero product property
  • Factoring
  • Root of a function
  • Zero of a function

Factoring Quadratic Equations Using the X Method

  • General form of a quadratic equation: ax2+bx+cax^2 + bx + c
  • Example:
    • Quadratic equation: x2+4x−5x^2 + 4x - 5
    • a = 1, b = 4, c = -5
Steps

  1. Write the product of a times c on the top of the x

    • a×c=1×−5=−5a \times c = 1 \times -5 = -5

  2. Write b on the bottom of the x

    • b = 4

  3. Find two factors that multiply to -5 and add to 4

    • Factors: 5 and -1 (since 5×−1=−55 \times -1 = -5 and 5+(−1)=45 + (-1) = 4)

  4. Rewrite the quadratic as four terms

    • x2+5x−1x−5x^2 + 5x - 1x - 5

  5. Group the terms for factoring

    • (x2+5x)+(−1x−5)(x^2 + 5x) + (-1x - 5)

  6. Factor out the common factor from each group

    • From (x2+5x)(x^2 + 5x), factor out x: x(x+5)x(x + 5)
    • From (−1x−5)(-1x - 5), factor out -1: −1(x+5)-1(x + 5)

  7. Rewrite the expression

    • x(x+5)−1(x+5)x(x + 5) - 1(x + 5)

  8. Factor out the common binomial factor

    • (x−1)(x+5)(x - 1)(x + 5)

  9. Check by multiplying the binomials

Practice

  • Use the x method to factor a different quadratic equation.