Question

which of the following is a solution to the quadratic equation 4x^2 + 2x = 10

Answer 1

To solve the quadratic equation \( 4x^2 + 2x = 10 \), we first need to set it equal to zero:

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

Next, 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 \( c = -10 \).

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

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

Next, we can substitute the values into the quadratic formula:

\[
x = \frac{-2 \pm \sqrt{164}}{2 \times 4} = \frac{-2 \pm \sqrt{164}}{8}.
\]

We can simplify \( \sqrt{164} \):

\[
\sqrt{164} = \sqrt{4 \times 41} = 2\sqrt{41}.
\]

Substituting this back into the formula gives:

\[
x = \frac{-2 \pm 2\sqrt{41}}{8} = \frac{-1 \pm \sqrt{41}}{4}.
\]

Thus, the two solutions to the equation \( 4x^2 + 2x = 10 \) are:

\[
x = \frac{-1 + \sqrt{41}}{4} \quad \text{and} \quad x = \frac{-1 - \sqrt{41}}{4}.
\]

These can be considered as the solutions to your quadratic equation. If you have specific options you want to compare against for the solution, please provide those options.