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### Unit Activity: Quadratic Relationships

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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 [tex]x[/tex] along the arch:
[tex]\[ -x^2 + 5x + 24 \][/tex]

#### Part A

Question:

Factor the expression, and use the factors to find the [tex]x[/tex]-intercepts of the quadratic relationship it represents.

Type the correct answer in each box, starting with the intercept with the lower value.

The [tex]x[/tex]-intercepts occur where [tex]x = \square[/tex] and [tex]x = \square[/tex].



Answer :

GinnyAnswer
To solve the problem, we need to factor the given quadratic expression and then use the factors to find the [tex]\( x \)[/tex]-intercepts.

The given quadratic expression is:
[tex]\[ -x^2 + 5x + 24 \][/tex]

### Step 1: Factor the Quadratic Expression
To factor the quadratic expression, we look for two binomials that multiply to give the original expression.

The factored form of the expression is:
[tex]\[ -(x - 8)(x + 3) \][/tex]

### Step 2: Find the [tex]\( x \)[/tex]-Intercepts
The [tex]\( x \)[/tex]-intercepts are found by setting each factor equal to zero and solving for [tex]\( x \)[/tex].

1. Set [tex]\( x - 8 = 0 \)[/tex]:
[tex]\[ x - 8 = 0 \implies x = 8 \][/tex]

2. Set [tex]\( x + 3 = 0 \)[/tex]:
[tex]\[ x + 3 = 0 \implies x = -3 \][/tex]

So, the [tex]\( x \)[/tex]-intercepts occur where [tex]\( x = -3 \)[/tex] and [tex]\( x = 8 \)[/tex].

### Conclusion
The [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = -3 \][/tex]
[tex]\[ x = 8 \][/tex]

These are the points where the height of the arch is zero, which represent the points along the coordinate plane where the arch meets the base level.

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