Question
Completing the Square & the Quadratic Formula Quick Check
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Question
Using the quadratic formula, find the solution to x2−6x−16=0.(1 point)
Responses
x=8, x=−2
x equals 8 , x equals negative 2
x=8, x=−8
x equals 8 , x equals negative 8
x=8
x equals 8
x=−2
Answers
Answered by
GPT-4o mini
To solve the equation \( x^2 - 6x - 16 = 0 \) using the quadratic formula, we start by identifying the coefficients \( a \), \( b \), and \( c \):
- \( a = 1 \)
- \( b = -6 \)
- \( c = -16 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, plug in the values of \( a \), \( b \), and \( c \):
1. Calculate the discriminant:
\[
b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100
\]
2. Substitute into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2(1)} = \frac{6 \pm 10}{2}
\]
3. Solve for the two possible values of \( x \):
- For the positive case:
\[
x = \frac{6 + 10}{2} = \frac{16}{2} = 8
\]
- For the negative case:
\[
x = \frac{6 - 10}{2} = \frac{-4}{2} = -2
\]
Thus, the solutions are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
The correct response is:
**x=8, x=−2** (or **x equals 8, x equals negative 2**)
- \( a = 1 \)
- \( b = -6 \)
- \( c = -16 \)
The quadratic formula is given by:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Now, plug in the values of \( a \), \( b \), and \( c \):
1. Calculate the discriminant:
\[
b^2 - 4ac = (-6)^2 - 4(1)(-16) = 36 + 64 = 100
\]
2. Substitute into the quadratic formula:
\[
x = \frac{-(-6) \pm \sqrt{100}}{2(1)} = \frac{6 \pm 10}{2}
\]
3. Solve for the two possible values of \( x \):
- For the positive case:
\[
x = \frac{6 + 10}{2} = \frac{16}{2} = 8
\]
- For the negative case:
\[
x = \frac{6 - 10}{2} = \frac{-4}{2} = -2
\]
Thus, the solutions are:
\[
x = 8 \quad \text{and} \quad x = -2
\]
The correct response is:
**x=8, x=−2** (or **x equals 8, x equals negative 2**)