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 \( x \):
- Calculating the first solution:
\[ x_1 = \frac{{-2 + 12.81}}{8} = \frac{10.81}{8} \approx 1.35 \]
- 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