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=−1.85
x equals negative 1.85

x=1.04
x equals 1.04

x=−0.77
x equals negative 0.77

x=1.15
x equals 1.15

1 answer

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:

  1. \(x = \frac{-2 + 12.81}{8} = \frac{10.81}{8} \approx 1.35\)
  2. \(x = \frac{-2 - 12.81}{8} = \frac{-14.81}{8} \approx -1.85\)

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.