Question

What are the x-intercepts of the function below?

g(x)=(4x−1)(x+2)


(1 point)
Responses

(-0.25, 0) and (2, 0)

(0.25, 0) and (-2, 0)

(1, 0) and (2, 0)

(-1, 0) and (2, 0)
Answer
What is the minimum of the graph described by y=(x+7)^2 +3
?(1 point)
Responses

(−7, −3)

(7, −3)

(−7, 3)

(7, 3)
Answer
What are the zeros of the graph of y = 4x^2 +9x −9
?(1 point)
Responses

3/4 and−3

−36 and 4

−3/4 and 3

4 and −9
Answer
Solve for x. Select all solutions.

x^2−25 = 0


(1 point)
Responses

5


-5


±5–√

0


25
Answer
Which of the following is a solution?

(x−3)^2−81=0


(1 point)
Responses

-12


3


12


-3
Answered by GPT-4o mini
To find the x-intercepts of the function \( g(x) = (4x - 1)(x + 2) \), we need to set \( g(x) \) equal to 0:

\[
(4x - 1)(x + 2) = 0
\]

This will hold true if either factor is equal to zero. Therefore, we can solve the equations:

1. \( 4x - 1 = 0 \)
2. \( x + 2 = 0 \)

For the first equation:

\[
4x - 1 = 0 \\
4x = 1 \\
x = \frac{1}{4} = 0.25
\]

For the second equation:

\[
x + 2 = 0 \\
x = -2
\]

Thus, the x-intercepts of the function are \( (0.25, 0) \) and \( (-2, 0) \).

The correct response is:

**(0.25, 0) and (-2, 0)**.
Answered by GPT-4o mini
The given function is \( y = (x + 7)^2 + 3 \).

This function is in the vertex form of a quadratic function, which is represented as \( y = a(x-h)^2 + k \), where the vertex is \( (h, k) \).

In this case:

- \( h = -7 \)
- \( k = 3 \)

Since the coefficient of the squared term is positive, the parabola opens upwards, indicating that the vertex represents the minimum point of the graph.

Thus, the minimum of the graph is at the vertex:

\[
(-7, 3)
\]

The correct response is:

**(−7, 3)**.
Answered by GPT-4o mini
To find the zeros of the quadratic function \( y = 4x^2 + 9x - 9 \), we need to set the equation equal to zero and solve for \( x \):

\[
4x^2 + 9x - 9 = 0
\]

We can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

where \( a = 4 \), \( b = 9 \), and \( c = -9 \).

First, we calculate the discriminant (\( b^2 - 4ac \)):

\[
b^2 - 4ac = 9^2 - 4 \cdot 4 \cdot (-9) = 81 + 144 = 225
\]

Now we can substitute back into the quadratic formula:

\[
x = \frac{-9 \pm \sqrt{225}}{2 \cdot 4} = \frac{-9 \pm 15}{8}
\]

Now we'll calculate the two possible values for \( x \):

1. For \( -9 + 15 = 6 \):

\[
x = \frac{6}{8} = \frac{3}{4}
\]

2. For \( -9 - 15 = -24 \):

\[
x = \frac{-24}{8} = -3
\]

Thus, the zeros of the function are \( \frac{3}{4} \) and \( -3 \).

The correct response is:

**3/4 and −3**.
Answered by GPT-4o mini
To solve the equation \( x^2 - 25 = 0 \), we can factor it as follows:

\[
x^2 - 25 = (x - 5)(x + 5) = 0
\]

Setting each factor equal to zero gives us:

1. \( x - 5 = 0 \) \(\Rightarrow x = 5\)
2. \( x + 5 = 0 \) \(\Rightarrow x = -5\)

So the solutions to the equation are \( x = 5 \) and \( x = -5 \).

Now, looking at the responses:

- **5** (correct)
- **-5** (correct)
- **±5–√** (incorrect)
- **0** (incorrect)
- **25** (incorrect)

Thus, the correct solutions are:

**5 and -5**.
Answered by GPT-4o mini
To solve the equation \( (x - 3)^2 - 81 = 0 \), we can start by isolating the squared term:

\[
(x - 3)^2 = 81
\]

Next, we take the square root of both sides:

\[
x - 3 = \pm 9
\]

Now, we solve for \( x \) in each case:

1. \( x - 3 = 9 \)
\[
x = 9 + 3 = 12
\]

2. \( x - 3 = -9 \)
\[
x = -9 + 3 = -6
\]

So the solutions to the equation are \( x = 12 \) and \( x = -6 \).

Now, looking at the provided responses:

- **-12** (incorrect)
- **3** (incorrect)
- **12** (correct)
- **-3** (incorrect)

Thus, the solution is:

**12**.