Use the graphing calculator to graph the quadratic function [tex]$y=x^2-5 x-36$[/tex]. Which value is a solution of [tex]$0=x^2-5 x$[/tex] - 36?
-9
-4
4
6
Answer (1)
Factor the quadratic equation x 2 − 5 x − 36 = 0 into ( x − 9 ) ( x + 4 ) = 0 .
Set each factor to zero: x − 9 = 0 or x + 4 = 0 .
Solve for x to find the solutions: x = 9 or x = − 4 .
From the given options, identify − 4 as a solution: − 4 .
Explanation
Understanding the Problem We are given the quadratic function y = x 2 − 5 x − 36 and asked to find which of the values − 9 , − 4 , 4 , 6 is a solution to the equation 0 = x 2 − 5 x − 36 . A solution to this equation is a value of x that makes the equation true, which corresponds to an x-intercept of the graph of the quadratic function.
Factoring the Quadratic We can solve the quadratic equation x 2 − 5 x − 36 = 0 by factoring. We are looking for two numbers that multiply to − 36 and add to − 5 . These numbers are − 9 and 4 . Thus, we can factor the quadratic as ( x − 9 ) ( x + 4 ) = 0 .
Finding the Solutions Setting each factor equal to zero, we have x − 9 = 0 or x + 4 = 0 . Solving for x , we get x = 9 or x = − 4 .
Checking the Solutions From the given options, − 4 is a solution. We can check this by substituting x = − 4 into the equation: ( − 4 ) 2 − 5 ( − 4 ) − 36 = 16 + 20 − 36 = 36 − 36 = 0 . Also, we can check x = 9 : ( 9 ) 2 − 5 ( 9 ) − 36 = 81 − 45 − 36 = 81 − 81 = 0 .
Final Answer Therefore, the value − 4 is a solution of 0 = x 2 − 5 x − 36 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a ball, determining the dimensions of a garden with a specific area, or modeling the profit of a business. For instance, if a company wants to maximize its profit based on the number of units sold, it might use a quadratic equation to model the relationship between the number of units and the profit. By finding the roots of the equation, the company can determine the sales levels at which it breaks even or achieves maximum profit. This helps in making informed business decisions.
