To solve the quadratic equation \( 4x^2 + 2x = 10 \), first, we can rearrange it to standard form:
\[ 4x^2 + 2x - 10 = 0 \]
Now, we can apply the quadratic formula, which is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this case, \( a = 4 \), \( b = 2 \), and \( c = -10 \).
Calculating the discriminant:
\[ b^2 - 4ac = 2^2 - 4 \cdot 4 \cdot (-10) = 4 + 160 = 164 \]
Now, we can use the quadratic formula to find the roots:
\[ x = \frac{-2 \pm \sqrt{164}}{2 \cdot 4} \]
Calculating \( \sqrt{164} \):
\[ \sqrt{164} \approx 12.81 \]
Substituting this back into the formula:
\[ x = \frac{-2 \pm 12.81}{8} \]
Calculating the two possible values for \( x \):
-
Using the positive root: \[ x = \frac{-2 + 12.81}{8} \approx \frac{10.81}{8} \approx 1.35 \]
-
Using the negative root: \[ x = \frac{-2 - 12.81}{8} \approx \frac{-14.81}{8} \approx -1.85 \]
Now, rounding to the nearest hundredth, the solutions are approximately:
- \( x \approx 1.35 \)
- \( x \approx -1.85 \)
From the given options, the solution that matches is:
\[ x = -1.85 \]
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