12.

Question

A rock is dropped from a bridge 320 feet above a river. The pathway that the rock takes can be modeled by the equation below. Approximately how long will it take the rock to reach the river?

h(t) = −16t2+320

(1 point)
Responses

2.5 seconds
2.5 seconds

3.5 seconds
3.5 seconds

3.8 seconds
3.8 seconds

4.5 seconds
4.5 seconds
Question 2
13.

Solve the equation. Simplify all irrational solutions.

5x2 +14x = x + 6

(1 point)
Responses

x={−3 , 25}
x={−3 , 25}

x={−3, 5}
x= {−3, 5}

x={15, 5}
x={15, 5}

x={23, 3}
x={23, 3}
Question 3
14.

Solve the equation. Simplify all irrational solutions.

x2 − 6x − 23 = 0

(1 point)
Responses

x = {4±32–√}
x = {4±32–√}

x = {3±42–√}
x = {3±42–√}

x = {±32−−√}
x = {±32−−√}

x = {−3±42–√}
x = {−3±42–√}
Question 4
15.

What is the value of c that completes the square trinomial to give the two solutions {17, -1}?

x2 − 16x − 13 = 4


(1 point)
Responses

17
17

81
81

64
64

8
8
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Answer 1

Let's solve each question step by step.

Question 1: Time for the rock to reach the river

The height of the rock as a function of time is given by: \[ h(t) = -16t^2 + 320 \] We want to find the time \( t \) when the rock reaches the river, which occurs when \( h(t) = 0 \): \[ -16t^2 + 320 = 0 \] To solve for \( t \):

1. Rearranging gives: \[ -16t^2 = -320 \]
2. Dividing both sides by -16: \[ t^2 = 20 \]
3. Taking the square root: \[ t = \sqrt{20} \approx 4.47 \] So, the time it takes approximately is 4.5 seconds.

Answer: 4.5 seconds

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Question 2: Solve the equation \( 5x^2 + 14x = x + 6 \)

1. Rearranging the equation gives: \[ 5x^2 + 14x - x - 6 = 0 \] \[ 5x^2 + 13x - 6 = 0 \]
2. We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
   • \( a = 5 \), \( b = 13 \), \( c = -6 \)
   • Discriminant: \[ b^2 - 4ac = 13^2 - 4 \cdot 5 \cdot (-6) = 169 + 120 = 289 \]
   • Roots: \[ x = \frac{-13 \pm \sqrt{289}}{10} \] \[ x = \frac{-13 \pm 17}{10} \]
   • Thus, \[ x_1 = \frac{4}{10} = 0.4 \] \[ x_2 = \frac{-30}{10} = -3 \]

From the answer choices, we find the solutions as: \[ x = {-3, 0.4} \]

Since \( 0.4 \) isn’t listed in the options, let’s check the given options closely: The closest option that matches the given solutions might require re-evaluation.

3. None of the provided responses seem accurate so the expected answer might be x = -3, 0.4 or something else, but it's not listed correctly.

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Question 3: Solve the equation \( x^2 - 6x - 23 = 0 \)

Again, using the quadratic formula:

1. Here \( a = 1 \), \( b = -6 \), \( c = -23 \)
2. The discriminant: \[ b^2 - 4ac = (-6)^2 - 4(1)(-23) = 36 + 92 = 128 \]
3. Roots: \[ x = \frac{6 \pm \sqrt{128}}{2 \cdot 1} \] \[ x = \frac{6 \pm 8\sqrt{2}}{2} \] \[ x = 3 \pm 4\sqrt{2} \]

Thus the correct answer is: x = {3 ± 4√2}

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Question 4: Completing the square

Given: \[ x^2 - 16x - 13 = 4 \] To complete the square:

1. Rearranging: \[ x^2 - 16x = 17 \]
2. Completing the square involves taking half of -16, squaring it:
   • Half of -16 is -8, and \((-8)^2 = 64\)
3. Thus, \[ (x - 8)^2 = 17 + 64 \] \[ (x - 8)^2 = 81 \]

Thus, the number that completes the square is: 64

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Final Answers Recap:

1. 4.5 seconds
2. The responses might require clarification since no option matches correctly.
3. x = {3 ± 4√2}
4. 64