Consider the function $f(x)=x^2+2 x-15$. What are the $x$-intercepts of the function?
Left-most $x$-intercept: ($\square$, 0)
Right-most $x$-intercept: ($\square$, 0)
Answer (1)
Find the roots of the quadratic equation x 2 + 2 x − 15 = 0 .
Factor the quadratic equation as ( x + 5 ) ( x − 3 ) = 0 .
Solve for x to find the x -intercepts: x = − 5 and x = 3 .
The left-most x -intercept is ( − 5 , 0 ) and the right-most x -intercept is ( 3 , 0 ) . Therefore, the x-intercepts are ( − 5 , 0 ) and ( 3 , 0 ) .
Explanation
Problem Analysis We are given the function f ( x ) = x 2 + 2 x − 15 and we want to find its x -intercepts. The x -intercepts are the points where the graph of the function intersects the x -axis, which means we need to find the values of x for which f ( x ) = 0 . In other words, we need to solve the quadratic equation x 2 + 2 x − 15 = 0 .
Factoring the Quadratic To solve the quadratic equation x 2 + 2 x − 15 = 0 , we can try to factor it. We are looking for two numbers that multiply to − 15 and add to 2 . These numbers are 5 and − 3 . So, we can factor the quadratic as ( x + 5 ) ( x − 3 ) = 0 .
Solving for x Now, we set each factor equal to zero and solve for x :
x + 5 = 0 or x − 3 = 0
Solving these equations, we get:
x = − 5 or x = 3
So, the x -intercepts are x = − 5 and x = 3 .
Identifying the Intercepts The x -intercepts are the points ( − 5 , 0 ) and ( 3 , 0 ) . The left-most x -intercept is ( − 5 , 0 ) and the right-most x -intercept is ( 3 , 0 ) .
Final Answer Therefore, the left-most x -intercept is ( − 5 , 0 ) and the right-most x -intercept is ( 3 , 0 ) .
Examples
Understanding x-intercepts is crucial in various real-world applications. For instance, if f ( x ) represents the profit of a company as a function of the number of units x sold, the x-intercepts (where f ( x ) = 0 ) indicate the break-even points, i.e., the number of units the company needs to sell to cover all costs. Similarly, in physics, if f ( x ) describes the height of a projectile as a function of horizontal distance x , the x-intercepts represent where the projectile lands. Knowing these points helps in making informed decisions in business and predicting outcomes in physics.
