%5E2%3D49-zero-product-property.jpeg "(x+2)^2=49 Zero Product Property")
Solving Quadratic Equations Using the Zero Product Property
The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero. This property is incredibly useful for solving quadratic equations. Let's explore how it works with the example:
(x + 2)^2 = 49
1. Rewrite the Equation in Standard Form
First, we need to rewrite the equation in standard form, which means setting it equal to zero:
- (x + 2)^2 - 49 = 0
2. Factor the Equation
Next, we can factor the equation. Notice that the left side of the equation is a difference of squares:
- (x + 2)^2 - 7^2 = 0
- (x + 2 + 7)(x + 2 - 7) = 0
3. Apply the Zero Product Property
We have now factored the equation into two factors. The Zero Product Property tells us that if the product of these factors is zero, then at least one of the factors must be zero:
- (x + 9) = 0 or (x - 5) = 0
4. Solve for x
Finally, we solve for x in each of the equations:
- x = -9 or x = 5
Therefore, the solutions to the equation (x + 2)^2 = 49 are x = -9 and x = 5.
In summary, to solve a quadratic equation using the Zero Product Property:
- Rewrite the equation in standard form (set it equal to zero).
- Factor the equation.
- Apply the Zero Product Property (set each factor equal to zero).
- Solve for x.
This method is a powerful tool for solving quadratic equations and can be used to solve various problems in mathematics, science, and engineering.