Use the zero product property to find the solutions to the equation $(x+2)(x+3)=12$.
A. $x=-1$ or $x=6$
B. $x=-3$ or $x=-2$
C. $x=-6$ or $x=1$
D. $x=2$ or $x=3$
Answer (2)
Expand the equation ( x + 2 ) ( x + 3 ) = 12 to get x 2 + 5 x + 6 = 12 .
Simplify the equation to x 2 + 5 x − 6 = 0 .
Factor the quadratic equation as ( x + 6 ) ( x − 1 ) = 0 .
Apply the zero product property to find the solutions x = − 6 or x = 1 .
x = − 6 or x = 1
Explanation
Rewrite the equation First, we need to rewrite the given equation ( x + 2 ) ( x + 3 ) = 12 in the standard quadratic form a x 2 + b x + c = 0 . To do this, we first expand the left side of the equation:
Expand the product Expanding ( x + 2 ) ( x + 3 ) , we get: x 2 + 3 x + 2 x + 6 = 12
Combine like terms Combining like terms, we have: x 2 + 5 x + 6 = 12
Set the equation to zero Now, subtract 12 from both sides to set the equation to zero: x 2 + 5 x + 6 − 12 = 0
Simplify the equation This simplifies to: x 2 + 5 x − 6 = 0
Factor the quadratic Next, we factor the quadratic equation x 2 + 5 x − 6 = 0 . We are looking for two numbers that multiply to -6 and add to 5. These numbers are 6 and -1. So, we can factor the equation as: ( x + 6 ) ( x − 1 ) = 0
Apply the zero product property Now, we apply the zero product property, which states that if ab = 0 , then a = 0 or b = 0 . In this case, we have: x + 6 = 0 or x − 1 = 0
Solve for x Solving for x in each equation, we get: x = − 6 or x = 1
Final solutions Therefore, the solutions to the equation ( x + 2 ) ( x + 3 ) = 12 are x = − 6 or x = 1 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For example, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. By solving these equations, they can determine the optimal dimensions and ensure the bridge's stability. Similarly, in physics, projectile motion can be described using quadratic equations, allowing us to predict the trajectory of a ball thrown in the air or the path of a rocket.
The solutions to the equation ( x + 2 ) ( x + 3 ) = 12 are x = − 6 or x = 1 . We derived this by expanding the expression, setting it to zero, factoring, and applying the zero product property. Thus, the correct answer is option C: x = − 6 or x = 1 .
;
