Which of the following is a solution to the quadratic equation 4x2+2x=10

Question

4
𝑥
2
+
2
𝑥
=
10
? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point)
Responses

x=1.04
𝑥
=
1.04
x equals 1.04

x=−0.77
𝑥
=
−
0.77
x equals negative 0.77

x=1.15
𝑥
=
1.15
x equals 1.15

x=−1.85

Answer 1

To solve the quadratic equation \(4x^2 + 2x = 10\), we first rearrange it 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 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 can substitute \(a\), \(b\), and the discriminant into the quadratic formula:

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

Calculating \(\sqrt{164}\):

\[ \sqrt{164} \approx 12.81 \text{ (rounded to two decimal places)} \]

Now substituting this into the formula gives:

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

Calculating both solutions:

1. For the \(+\) case:

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

2. For the \(−\) case:

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

Now we round both solutions to the nearest hundredth:

• The first solution is approximately \(1.35\).
• The second solution is approximately \(-1.85\).

Given the options, the correct answer is:

x = -1.85.