Question

Graph the quadratic y-x^2+3x+8. What is the x-intercept(s)?
a. 1.5
b. There is no x-intercept
c. (-1.5, 5.75)
d. (0, 8)

Answer 1

To find the x-intercepts of the quadratic equation \( y = -x^2 + 3x + 8 \), we need to set \( y = 0 \) and solve for \( x \):

\[
0 = -x^2 + 3x + 8
\]

Rearranging it gives us:

\[
x^2 - 3x - 8 = 0
\]

We can solve this quadratic equation using the quadratic formula:

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

where \( a = 1 \), \( b = -3 \), and \( c = -8 \).

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

\[
(-3)^2 - 4(1)(-8) = 9 + 32 = 41
\]

Now we substitute back into the quadratic formula:

\[
x = \frac{-(-3) \pm \sqrt{41}}{2(1)} = \frac{3 \pm \sqrt{41}}{2}
\]

Thus, the two x-intercepts are:

\[
x_1 = \frac{3 + \sqrt{41}}{2}
\]

\[
x_2 = \frac{3 - \sqrt{41}}{2}
\]

Calculating these values:

1. \( \sqrt{41} \approx 6.4 \), so:
- \( x_1 \approx \frac{3 + 6.4}{2} = \frac{9.4}{2} \approx 4.7 \)
- \( x_2 \approx \frac{3 - 6.4}{2} = \frac{-3.4}{2} \approx -1.7 \)

The approximate x-intercepts are \( x_1 \approx 4.7 \) and \( x_2 \approx -1.7 \).

Now, looking at the provided options for x-intercepts:

1. \( a. 1.5 \)
2. \( b. There is no x-intercept \)
3. \( c. (-1.5, 5.75) \)
4. \( d. (0, 8) \)

From our calculation, we found the x-intercepts to be approximately \( -1.7 \) and \( 4.7 \). This means that the x-intercepts we found do not directly match any of the given options.

However, we can informally conclude that:

- **The x-intercepts do exist** and are numerical values (roughly \( -1.7 \) and \( 4.7 \)), which indicates that option **b** ("There is no x-intercept") is incorrect.

- None of the other options provides an accurate representation based on our calculations.

Thus, the answer is there are x-intercepts, but none of the options match correctly. The correct answer should include the x-intercepts \( x \approx -1.7 \) and \( x \approx 4.7 \).