Question

Use the quadratic formula to solve the following quadratic equation:

−4x2+16x−4=8
(1 point)
Responses

x=−0.24, x=4.24
x=−0.24, x=4.24

x=−1, x=−3
x=−1, x=−3

x=−4.24, x=0.24
x=−4.24, x=0.24

x=1, x=3

Answer 1

To solve the quadratic equation \(-4x^2 + 16x - 4 = 8\), we first need to rearrange it into standard form, \(ax^2 + bx + c = 0\).

Start by subtracting 8 from both sides:

\[
-4x^2 + 16x - 4 - 8 = 0 \\
-4x^2 + 16x - 12 = 0
\]

Now, we can divide the entire equation by -4 to simplify it:

\[
x^2 - 4x + 3 = 0
\]

Now we can apply the quadratic formula, which is:

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

In our equation, \(a = 1\), \(b = -4\), and \(c = 3\).

Let's calculate the discriminant \(b^2 - 4ac\):

\[
b^2 - 4ac = (-4)^2 - 4(1)(3) = 16 - 12 = 4
\]

Now we can substitute the values into the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{4}}{2(1)} \\
x = \frac{4 \pm 2}{2}
\]

This gives us two solutions:

1. \(x = \frac{4 + 2}{2} = \frac{6}{2} = 3\)
2. \(x = \frac{4 - 2}{2} = \frac{2}{2} = 1\)

Thus, the solutions to the quadratic equation are:

\[
x = 1 \quad \text{and} \quad x = 3
\]

Therefore, the correct response is:

**x=1, x=3**