To find the solutions to the quadratic equation \(4x^2 + 2x = 10\), we first rearrange the equation to standard form:
\(4x^2 + 2x - 10 = 0\).
Next, we can use the quadratic formula, which is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \(a = 4\), \(b = 2\), and \(c = -10\). Now, we will compute the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 4 \cdot (-10) = 4 + 160 = 164 \]
Now substituting into the quadratic formula:
\[ x = \frac{-2 \pm \sqrt{164}}{2 \cdot 4} = \frac{-2 \pm \sqrt{164}}{8} \]
Calculating \(\sqrt{164}\):
\(\sqrt{164} \approx 12.81\) (rounded to two decimal places).
Now substituting this back into the equation:
\[ x = \frac{-2 \pm 12.81}{8} \]
This gives us two solutions:
- \(x = \frac{-2 + 12.81}{8} = \frac{10.81}{8} \approx 1.35\)
- \(x = \frac{-2 - 12.81}{8} = \frac{-14.81}{8} \approx -1.85\)
Thus, rounding to the nearest hundredth, our solutions are approximately \(x \approx -1.85\) and \(x \approx 1.35\).
From the options given, the only correct solution is:
x = -1.85.