The quadratic function $y=-x^2+10 x-8$ models the height of a trestle on a bridge. The $x$-axis represents ground level.
To find where the section of the bridge meets ground level, solve $0=-x^2+10 x-8$
Where does this section of the bridge meet ground level?
Choose an equation that would be used to solve
$0=-x^2+10 x-8$
$(x+25)^2=-8$
$(x-5)^2=17$
$(x-10)^2=25
Answer (1)
Rewrite the equation 0 = − x 2 + 10 x − 8 as x 2 − 10 x = − 8 .
Complete the square by adding ( − 10/2 ) 2 = 25 to both sides: x 2 − 10 x + 25 = − 8 + 25 .
Rewrite the left side as a squared term: ( x − 5 ) 2 = 17 .
The equivalent equation is ( x − 5 ) 2 = 17 .
Explanation
Understanding the Problem We are given the quadratic function y = − x 2 + 10 x − 8 which models the height of a trestle on a bridge. The x -axis represents ground level, and we want to find where the bridge meets ground level. This means we need to solve the equation 0 = − x 2 + 10 x − 8 for x . We are asked to choose the correct equivalent equation from the given options.
Rewriting the Equation To solve the quadratic equation 0 = − x 2 + 10 x − 8 , we can complete the square. First, let's rewrite the equation as x 2 − 10 x = − 8 .
Completing the Square Now, we complete the square. We take half of the coefficient of the x term, which is − 10 , so half of it is − 5 . We square − 5 to get ( − 5 ) 2 = 25 . We add 25 to both sides of the equation:
Adding to Both Sides x 2 − 10 x + 25 = − 8 + 25
Rewriting as a Squared Term Now we can rewrite the left side as a squared term:
Final Equation ( x − 5 ) 2 = 17
The Answer Therefore, the correct equation is ( x − 5 ) 2 = 17 .
Examples
Understanding quadratic functions and their roots is crucial in various real-world applications. For instance, when designing a bridge, engineers use quadratic equations to model the arch's shape. By finding the roots of the equation, they can determine where the arch meets the ground, ensuring the bridge's stability and proper construction. Similarly, in projectile motion, quadratic equations help calculate the trajectory of an object, such as a ball thrown in the air, predicting its landing point.
