Using the quadratic formula, find the solution to 4x2+4x−15=0.(1 point)

Question

Using the quadratic formula, find the solution to 4x2+4x−15=0.(1 point)
Responses

x=−2.5
x is equal to negative 2 point 5 - no response given

x=1.5
x is equal to 1 point 5 - incorrect

x=−1.5, x=−2.5
x is equal to negative 1 point 5, x is equal to negative 2 point 5 - no response given

x=1.5, x=−2.5

GPT-4o mini · Answer

To solve the quadratic equation $4x^2 + 4x - 15 = 0$ using the quadratic formula, we use: $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Here, $a = 4$, $b = 4$, and $c = -15$. 1. **Calculate the discriminant $b^2 - 4ac$**: $$ b^2 - 4ac = 4^2 - 4(4)(-15) = 16 + 240 = 256 $$ 2. **Calculate the square root of the discriminant**: $$ \sqrt{256} = 16 $$ 3. **Substitute $b$, the square root of the discriminant, and $a$ into the quadratic formula**: $$ x = \frac{-4 \pm 16}{2(4)} = \frac{-4 \pm 16}{8} $$ 4. **Calculate the two possible values for $x$**: - For the positive case ($+$): $$ x = \frac{-4 + 16}{8} = \frac{12}{8} = 1.5 $$ - For the negative case ($-$): $$ x = \frac{-4 - 16}{8} = \frac{-20}{8} = -2.5 $$ Thus, the solutions are: $$ x = 1.5 \quad \text{and} \quad x = -2.5 $$ Therefore, the correct response is: **x=1.5, x=−2.5**.