Use factoring to solve the quadratic equation. Check by substitution or by using a graphing utility and identifying $x$-intercepts.
[tex]x^2-x-72=0[/tex]
Rewrite the equation in factored form.
[tex]\square[/tex] $=0$
(Factor completely.)
The solution set is [tex]\square[/tex].
(Use a comma to separate answers as needed. Type each solution only once.)
Answer (2)
Factor the quadratic expression x 2 − x − 72 to ( x − 9 ) ( x + 8 ) .
Set each factor equal to zero: x − 9 = 0 and x + 8 = 0 .
Solve for x : x = 9 and x = − 8 .
The solution set is − 8 , 9 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − x − 72 = 0 . Our goal is to solve this equation by factoring. This involves rewriting the quadratic expression as a product of two binomials.
Finding the Factors To factor the quadratic expression x 2 − x − 72 , we need to find two numbers that multiply to -72 and add up to -1. Let's think of factor pairs of 72: 1 and 72, 2 and 36, 3 and 24, 4 and 18, 6 and 12, 8 and 9. Since the product is negative, one of the numbers must be negative. Since the sum is -1, the larger number must be negative. The pair that works is 8 and -9 because 8 × − 9 = − 72 and 8 + ( − 9 ) = − 1 .
Factoring the Quadratic Expression Now we can rewrite the quadratic expression in factored form using the numbers 8 and -9: ( x + 8 ) ( x − 9 ) = 0 .
Setting Each Factor to Zero To find the solutions for x , we set each factor equal to zero: x + 8 = 0 and x − 9 = 0 .
Solving for x Solving for x in each equation: For x + 8 = 0 , we subtract 8 from both sides to get x = − 8 . For x − 9 = 0 , we add 9 to both sides to get x = 9 .
Stating the Solution Set Therefore, the solution set is { − 8 , 9 } .
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a ball, determining the dimensions of a garden with a specific area, or modeling the growth of a population. Factoring is a method to solve these equations, providing a way to find the values that satisfy the given conditions. For example, if you want to build a rectangular garden with an area of 72 square feet and the length should be 1 foot more than the width, you can use a quadratic equation to find the dimensions of the garden.
To solve the quadratic equation x 2 − x − 72 = 0 , we factor it into ( x − 9 ) ( x + 8 ) = 0 and find the solutions x = 9 and x = − 8 . The solution set is { − 8 , 9 } .
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