Using the quadratic formula, find the solution to 4x2+4x−15=0

Question

.(1 point)
Responses

x=1.5
x equals 1.5

x=−1.5
, x=−2.5
x equals negative 1.5 , x equals negative 2.5

x=−2.5
x equals negative 2.5

x=1.5
, x=−2.5

Answer 1

To solve the quadratic equation \( 4x^2 + 4x - 15 = 0 \) using the quadratic formula, we use the formula:

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

For the equation \( 4x^2 + 4x - 15 = 0 \), the coefficients are:

• \( a = 4 \)
• \( b = 4 \)
• \( c = -15 \)

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

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

2. Now, we can apply the quadratic formula:

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

3. Calculate \( \sqrt{256} \):

\[ \sqrt{256} = 16 \]

4. Substitute back into the formula:

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

5. Now calculate the two possible solutions:

• For the \( + \) case:

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

• For the \( - \) case:

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

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

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

Therefore, the correct response is:

x=1.5, x=−2.5.