It may appear that there is only one factor in the next example. Remember, however, that
means
.
Solve:
.
Solution
Rewrite the left side as a product.
Use the Zero Product Property and
set each factor to 0.
Solve the equations.
When a solution repeats, we call it
a double root.
Check your answer.
This image shows the steps for solving the equation (y – 8) squared = 0. The first step is to write the left hand side as a product, (y – 8)(y – 8) = 0. The next step is using the zero product property and set each factor equal to 0, y – 8 = 0 and y – 8 -= 0. Solve both equations, y = 8 and y = 8. When the solution repeats, it is a double root. Finally, check the solution by substituting back into the original equation.
Each of the equations we have solved in this section so far had one side in factored form. In order to use the Zero Product Property, the quadratic equation must be factored, with zero on one side. So we be sure to start with the quadratic equation in standard form,
. Then we factor the expression on the left.
Before we factor, we must make sure the quadratic equation is in standard form.
Solve:
.
Solution
Write the quadratic equation in standard form.
Factor the quadratic expression.
Use the Zero Product Property
to set each factor to 0.
Solve each equation.
Check your answers.
This image shows the steps for solving the equation 2 y squared = 13 y + 45. The first step is writing the equation in standard quadratic form, 2 y squared – 13 y – 45 = 0. The second step is to factor the quadratic expression, (2 y + 5)(y – 9) = 0. The third step is to use the zero product property to set each factor equal to 0, 2 y + 5 = 0 or y – 9 = 0. Solve each equation, y = −5/2 or y = 9. Finally, check the answers by substituting them back into the original equation.
Use the Zero Product Property
to set each factor to 0.
Solve each equation.
Check your answers.
This image shows the steps for solving the equation 5 x squared – 13 x = 7 x. The first step is writing the equation in standard quadratic form, 5 x squared – 20 x = 0. The second step is to factor the quadratic expression, 5 x(x – 4)= 0. The third step is to use the zero product property to set each factor equal to 0, 5 x = 0 or x − 4 = 0. Solve each equation, x = 0 or x = 4. Finally, check the answers by substituting them back into the original equation.
