What values of the variables make each of the following equations true?

- z-7=0
- 13a=0
- 3(d+4)=0
- x\cdot y=0

The **zero product property ** states that if a product of two or more factors is equal to 0, then at least one of the factors must be equal to 0. That is, if we know that xy=0 then at least one of x=0 or \\y=0 must be true.

We can use this property to solve quadratic equations by first writing the equation in **factored form**: a\left(x-x_1\right)\left(x-x_2\right)=0

If we can write a quadratic equation in the **factored form**, then we know that either x-x_1=0 or \\ x-x_2=0. This means that the solutions to the quadratic equation are x=x_1 and x=x_2. This approach can be useful if the equation has rational solutions.

Given a quadratic function f(x), the following statements are equivalent for any real number, k, such that f(k)=0:

k is a zero of the function f(x), located at (k,0)

(x-k) is a factor of f(x)

k is a solution or root of the equation f(x)=0

the point (k,0) is an

**x-intercept**for the graph of y=f(x)

Let's take a look at this for a specific function:

Solve the following equations by factoring:

a

x^2+6x-55=0

Worked Solution

b

3x^2+3x-10=8

Worked Solution

Luis throws a ball straight into the air. The path of the ball can be modeled by the equation {y=-5x^2+14x+3} where x represents the time the ball is in the air in seconds and y represents the height of the ball in meters. How long will it take the ball to hit the ground?

Worked Solution

Idea summary

We can use the zero product property to solve quadratic equations by first writing the equation in factored form: a\left(x-x_1\right)\left(x-x_2\right)=0

then setting each factor equal to zero and solving for x.