Find the zeros of the quadratic function: [tex]$f(x)=x^2+10 x+24$[/tex]
A. [tex]$x=4$[/tex] [tex]$x=6$[/tex]
B. [tex]$x=-8$[/tex] [tex]$x=-3$[/tex]
C. [tex]$x=8$[/tex] [tex]$x=3$[/tex]
D. [tex]$x=-4$[/tex] [tex]$x=-6$[/tex]
Answer (1)
Factor the quadratic expression: x 2 + 10 x + 24 = ( x + 4 ) ( x + 6 ) .
Set each factor equal to zero: x + 4 = 0 or x + 6 = 0 .
Solve for x : x = − 4 or x = − 6 .
The zeros of the quadratic function are x = − 4 , x = − 6 .
Explanation
Understanding the Problem We are given the quadratic function f ( x ) = x 2 + 10 x + 24 and we want to find its zeros. The zeros of a function are the values of x for which f ( x ) = 0 . In other words, we need to solve the equation x 2 + 10 x + 24 = 0 .
Factoring the Quadratic To solve the quadratic equation x 2 + 10 x + 24 = 0 , we can try to factor the quadratic expression. We are looking for two numbers that multiply to 24 and add up to 10. These numbers are 6 and 4, since 6 × 4 = 24 and 6 + 4 = 10 . Therefore, we can write the quadratic expression as ( x + 6 ) ( x + 4 ) .
Setting Factors to Zero Now we have the equation ( x + 6 ) ( x + 4 ) = 0 . For this equation to be true, at least one of the factors must be equal to zero. So we have two possibilities:
x + 6 = 0
x + 4 = 0
Solving for x Solving the first equation, x + 6 = 0 , we subtract 6 from both sides to get x = − 6 .
Solving the second equation, x + 4 = 0 , we subtract 4 from both sides to get x = − 4 .
Therefore, the zeros of the quadratic function are x = − 6 and x = − 4 .
Final Answer The zeros of the quadratic function f ( x ) = x 2 + 10 x + 24 are x = − 6 and x = − 4 .
Examples
Imagine you are designing a rectangular garden and you know the area needs to be 24 square meters and the total length of fencing required is 20 meters. Finding the zeros of the quadratic equation x 2 − 10 x + 24 = 0 (where x represents the length of one side) helps you determine the possible dimensions of the garden. This ensures you meet both the area and perimeter requirements efficiently. Understanding quadratic equations is crucial in optimizing designs and resource allocation in various practical scenarios.
