Chapter 9.4, Problem 54HP

To determine

Solve the quadratic function by using Zero Product Property.

Answer to Problem 54HP

  $y=4a-2b+c$ or $y=9a-3b+c$

Explanation of Solution

Given Information: Quadratic function by using Zero Product Property.

The general equation $y=a{x}^2+bx+c$

Calculation:

The Zero Product Property states that if ab = 0, then either a = 0 or b = 0, or both a and b are 0. When the product of factors equals zero, one or more of the factors must also equal zero. Once the polynomial is factored, set each factor equal to zero and solve them separately.

  $(x+2)(x+3)=0$

Now solve each factor by setting each one equal to zero and solving the resulting linear equations:

  $(x+2)=0$ or $(x+3)=0$

Now,

  $x=-2$ or $x=-3$

These two values are the solution to the original quadratic equation. So

  $y=a\times {\left(-2\right)}^2+b\times \left(-2\right)+c$ or $y=a\times {\left(-3\right)}^2+b\times \left(-3\right)+c$

Hence,

  $y=4a-2b+c$ or $y=9a-3b+c$