Question

Using the quadratic formula, find the solution to 4x2+4x−15=0 .(1 point) Responses x=1.5 x is equal to 1 point 5 - no response given x=−1.5 , x=−2.5 x is equal to negative 1 point 5, x is equal to negative 2 point 5 - incorrect x=−2.5 x is equal to negative 2 point 5 - no response given x=1.5 , x=−2.5

Answer 1

To solve the quadratic equation \(4x^2 + 4x - 15 = 0\) using the quadratic formula, we start with the standard form of a quadratic equation:

\[
ax^2 + bx + c = 0
\]

Here, \(a = 4\), \(b = 4\), and \(c = -15\).

The quadratic formula is given by:

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

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

\[
b^2 = 4^2 = 16
\]
\[
4ac = 4 \cdot 4 \cdot (-15) = -240
\]
\[
b^2 - 4ac = 16 + 240 = 256
\]

Now we can substitute these values into the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{256}}{2 \cdot 4}
\]

Calculating \(\sqrt{256}\):

\[
\sqrt{256} = 16
\]

Now plug this back into the formula:

\[
x = \frac{-4 \pm 16}{8}
\]

This gives us two possible solutions:

1. Using the plus sign:

\[
x = \frac{-4 + 16}{8} = \frac{12}{8} = \frac{3}{2} = 1.5
\]

2. Using the minus sign:

\[
x = \frac{-4 - 16}{8} = \frac{-20}{8} = -\frac{5}{2} = -2.5
\]

Thus, the solutions to the equation \(4x^2 + 4x - 15 = 0\) are:

\[
x = 1.5 \quad \text{and} \quad x = -2.5
\]

The correct response is: **\(x = 1.5\), \(x = -2.5\)**.