Solve the quadratic equation [tex]x^2+x=72[/tex] by factoring.
A) [tex]x=-8, x=9[/tex]
B) [tex]x=8, x=9[/tex]
C) [tex]x=-9, x=8[/tex]
D) [tex]x=-9, x=-8[/tex]
Answer (1)
Rewrite the equation in standard form: x 2 + x − 72 = 0 .
Factor the quadratic expression: ( x + 9 ) ( x − 8 ) = 0 .
Set each factor to zero and solve for x : x + 9 = 0 or x − 8 = 0 .
The solutions are x = − 9 and x = 8 , so the answer is x = − 9 , x = 8 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + x = 72 and asked to solve it by factoring. Our goal is to find the values of x that satisfy this equation.
Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . To do this, we subtract 72 from both sides of the equation: x 2 + x − 72 = 0
Factoring the Quadratic Expression Now, we need to factor the quadratic expression x 2 + x − 72 . We are looking for two numbers that multiply to -72 and add to 1. These numbers are 9 and -8, since 9 × − 8 = − 72 and 9 + ( − 8 ) = 1 . Therefore, we can factor the quadratic expression as follows: ( x + 9 ) ( x − 8 ) = 0
Solving for x To find the solutions for x , we set each factor equal to zero and solve for x :
x + 9 = 0 Subtract 9 from both sides: x = − 9
x − 8 = 0 Add 8 to both sides: x = 8
So, the solutions are x = − 9 and x = 8 .
Final Answer Comparing our solutions with the given options, we see that option C matches our solutions: x = − 9 , x = 8 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For instance, if you want to build a rectangular garden with an area of 72 square meters and the length needs to be one meter longer than the width, you can model this situation with the quadratic equation x 2 + x = 72 , where x represents the width of the garden. Solving this equation helps you find the dimensions that satisfy the given conditions.
