Question

task 1
robotics competition
in this activity, you will use factoring and graphing to solve real - world problems. you will also describe transformations to the graph of (y = x^{2}).
question 1
jen and her friends are designing a robot for a stem competition. the goal of the competition is to have a robot complete an obstacle course in the least amount of time.
jen and her friends need to consider the robot’s height and its width at its base. the robot must be short enough and narrow enough to navigate through several arches along the obstacle course. the director of the competition laid out components of the obstacle course on a coordinate plane. instead of giving the width and maximum height of the arch, the director created an expression based on the location of the arch on the coordinate plane and gave each team this expression to represent the height of the arch, in inches, at any value of (x) along the arch: (-x^{2}+5x + 24).
part a
factor the expression, and use the factors to find the (x) - intercepts of the quadratic relationship it represents.
type the correct answer in each box, starting with the intercept with the lower value.
the (x) - intercepts occur where (x=square) and (x=square).

Explanation:

Step1: Factor the quadratic expression

We have the quadratic expression \(-x^{2}+5x + 24\). First, multiply both sides by - 1 to make the coefficient of \(x^{2}\) positive: \(x^{2}-5x - 24\).
We need to find two numbers that multiply to \(-24\) and add up to \(-5\). The numbers are \(-8\) and \(3\) since \((-8)\times3=-24\) and \(-8 + 3=-5\).
So, \(x^{2}-5x - 24=(x - 8)(x+3)\). But we had multiplied by - 1 earlier, so \(-x^{2}+5x + 24=-(x - 8)(x + 3)=( - x + 8)(x+3)=(8 - x)(x + 3)\) or we can also factor it as \(-(x^{2}-5x - 24)=-(x - 8)(x + 3)\)

Step2: Find the x - intercepts

To find the x - intercepts, we set \(y = 0\) (since the height \(y=-x^{2}+5x + 24\) and x - intercepts are where \(y = 0\)).
Set \(-x^{2}+5x + 24=0\)
Multiply both sides by - 1: \(x^{2}-5x - 24=0\)
Using the factored form \((x - 8)(x + 3)=0\) (from the positive coefficient quadratic we found earlier)
We set each factor equal to zero:
\(x-8 = 0\) gives \(x = 8\)
\(x + 3=0\) gives \(x=-3\)

Answer:

The x - intercepts occur where \(x=-3\) and \(x = 8\)