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 $0=-x^2+10 x-8$
$(x+25)^2=-8$
$(x-5)^2=17$
$(x-10)^2=25$
Solve the equation to find where the trestle meets ground level. Enter your answers from least to greatest and round to the nearest tenth.
The trestle meets the ground at $\square$ units and $\square$ units.
Answer (1)
Rewrite the equation 0 = − x 2 + 10 x − 8 as x 2 − 10 x + 8 = 0 .
Complete the square: ( x − 5 ) 2 = 17 .
Solve for x: x = 5 ± 17 .
Approximate the solutions to the nearest tenth: The trestle meets the ground at 0.9 units and 9.1 units.
Explanation
Understanding the Problem The problem states that the height of a trestle on a bridge is modeled by the quadratic function y = − x 2 + 10 x − 8 , where the x-axis represents ground level. We need to find where the trestle meets the ground, which means we need to solve the equation 0 = − x 2 + 10 x − 8 for x .
Rewriting the Equation First, let's rewrite the equation 0 = − x 2 + 10 x − 8 as x 2 − 10 x + 8 = 0 . This makes it easier to complete the square.
Completing the Square To complete the square, we take half of the coefficient of the x term, which is − 10 , and square it: ( 2 − 10 ) 2 = ( − 5 ) 2 = 25 . We add this to both sides of the equation x 2 − 10 x = − 8 . So, we have x 2 − 10 x + 25 = − 8 + 25 , which simplifies to ( x − 5 ) 2 = 17 .
Solving for x Now, we solve for x by taking the square root of both sides: x − 5 = ± 17 . This gives us two possible solutions: x = 5 + 17 and x = 5 − 17 .
Approximating the Solutions We need to approximate these values to the nearest tenth. We know that 17 is approximately 4.123. Therefore, x 1 = 5 − 17 ≈ 5 − 4.123 = 0.877 and x 2 = 5 + 17 ≈ 5 + 4.123 = 9.123 . Rounding to the nearest tenth, we get x 1 ≈ 0.9 and x 2 ≈ 9.1 .
Final Answer The trestle meets the ground at approximately 0.9 units and 9.1 units.
Examples
Imagine you are designing a bridge and need to know where the arch of the bridge will meet the ground. The quadratic equation helps you model the arch, and solving it tells you the exact points where the bridge touches the ground on either side. This is crucial for planning the foundations and ensuring the bridge is stable and safe. By understanding the roots of the quadratic equation, engineers can accurately determine these contact points, making the bridge construction precise and reliable.
