Solve $z^2-15 z+56=0$ using the zero product property.
A) $z=-4,7$
B) $z=8,7$
C) $z=8,-4$
D) $z=-5,-2$
Answer (1)
Factor the quadratic equation: z 2 − 15 z + 56 = ( z − 7 ) ( z − 8 ) .
Apply the zero product property: ( z − 7 ) ( z − 8 ) = 0 implies z − 7 = 0 or z − 8 = 0 .
Solve for z : z = 7 or z = 8 .
The solutions are z = 8 , 7 .
Explanation
Understanding the Problem We are given the quadratic equation z 2 − 15 z + 56 = 0 and asked to solve it using the zero product property. This means we need to factor the quadratic expression into two binomials.
Factoring the Quadratic We need to find two numbers that multiply to 56 and add up to 15. Let's list the factor pairs of 56:
1 and 56 2 and 28 4 and 14 7 and 8
The pair 7 and 8 add up to 15, so we can use these numbers to factor the quadratic.
Applying the Zero Product Property We can write the quadratic expression as ( z − 7 ) ( z − 8 ) = 0 .
Solving for z Now, we apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. So, we have two equations:
z − 7 = 0 or z − 8 = 0
Solving for z in each equation, we get:
z = 7 or z = 8
Finding the Solution Therefore, the solutions to the quadratic equation are z = 7 and z = 8 . Comparing these solutions with the given options, we see that option B matches our solutions.
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and perimeter, or modeling growth and decay processes. For example, if you're launching a rocket, you can use a quadratic equation to model its height as a function of time. By solving the equation, you can determine when the rocket will reach a certain height or when it will land back on the ground. This helps in planning and executing the launch effectively.
