Question

Which of the following is a solution to the quadratic equation 4x2+2x=10
? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=1.15
x equals 1.15

x=−0.77
x equals negative 0.77

x=1.04
x equals 1.04

x=−1.85

Answer 1

To find the solutions to the quadratic equation \( 4x^2 + 2x = 10 \), we first rearrange the equation to standard form:

\[
4x^2 + 2x - 10 = 0
\]

Now we can apply the quadratic formula, which is given by:

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

In our equation, \( a = 4 \), \( b = 2 \), and \( c = -10 \).

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

\[
b^2 - 4ac = 2^2 - 4(4)(-10) = 4 + 160 = 164
\]

Now, we apply the quadratic formula:

\[
x = \frac{{-2 \pm \sqrt{164}}}{2(4)} = \frac{{-2 \pm \sqrt{164}}}{8}
\]

Next, we simplify \( \sqrt{164} \):

\[
\sqrt{164} \approx 12.81 \quad \text{(since } 12.81^2 \approx 164\text{)}
\]

Now, substitute back into the formula:

\[
x = \frac{{-2 \pm 12.81}}{8}
\]

Calculating the two possible values for \( x \):

1. Calculating the first solution:

\[
x_1 = \frac{{-2 + 12.81}}{8} = \frac{10.81}{8} \approx 1.35
\]

2. Calculating the second solution:

\[
x_2 = \frac{{-2 - 12.81}}{8} = \frac{-14.81}{8} \approx -1.85
\]

After rounding, we find the solutions to be approximately:

- \( x \approx 1.35 \)
- \( x \approx -1.85 \)

Among the options provided, the matching solution is:

**x = -1.85**