Question

Solve for x: . (1 point)
=2+ 4x
x−3
12
x−3
7
2
9
2
no solutions
2

Answer 1

It seems like you're trying to set an equality involving \( x \). To clarify, let's rewrite the expression. You may have written it as:

\[
\frac{1}{x - 3} = 2 + 4x
\]

To solve for \( x \), we can first clear the fraction by multiplying both sides by \( x - 3 \) (assuming \( x \neq 3 \)):

\[
1 = (2 + 4x)(x - 3)
\]

Now, distribute on the right side:

\[
1 = 2x - 6 + 4x^2 - 12x
\]

Combine like terms:

\[
1 = 4x^2 - 10x - 6
\]

Now set the equation to zero:

\[
4x^2 - 10x - 7 = 0
\]

Next, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 4 \), \( b = -10 \), \( c = -7 \):

Calculate the discriminant:

\[
b^2 - 4ac = (-10)^2 - 4(4)(-7) = 100 + 112 = 212
\]

Now, substitute back into the quadratic formula:

\[
x = \frac{10 \pm \sqrt{212}}{8}
\]

Simplifying further, \( \sqrt{212} = \sqrt{4 \cdot 53} = 2\sqrt{53} \):

\[
x = \frac{10 \pm 2\sqrt{53}}{8} = \frac{5 \pm \sqrt{53}}{4}
\]

Thus, there are two solutions for \( x \):

\[
x = \frac{5 + \sqrt{53}}{4} \quad \text{and} \quad x = \frac{5 - \sqrt{53}}{4}
\]

If you meant something different, please clarify the equation, and I can help further!