Question

What is the positive solution to this equation?



`4x^{2}+12x\ =\ 135`

Answer 1

To solve the equation \( 4x^2 + 12x = 135 \), we start by rearranging it into standard form. Subtract \( 135 \) from both sides:

\[
4x^2 + 12x - 135 = 0
\]

Next, we can simplify the equation by dividing all terms by \( 3 \):

\[
\frac{4x^2}{3} + \frac{12x}{3} - \frac{135}{3} = 0
\]

This simplifies to:

\[
\frac{4}{3}x^2 + 4x - 45 = 0
\]

Now, we will multiply the entire equation by \( 3 \) to eliminate the fraction:

\[
4x^2 + 12x - 135 = 0
\]

Now we can use the quadratic formula to solve this, which is:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \( a = 4 \), \( b = 12 \), and \( c = -135 \).

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

\[
b^2 = 12^2 = 144
\]
\[
4ac = 4 \cdot 4 \cdot (-135) = -2160
\]
\[
b^2 - 4ac = 144 + 2160 = 2304
\]

Now we can take the square root of the discriminant:

\[
\sqrt{2304} = 48
\]

Now substitute back into the quadratic formula:

\[
x = \frac{-12 \pm 48}{2 \cdot 4}
\]
\[
x = \frac{-12 \pm 48}{8}
\]

Now we calculate the two possible values for \( x \):

1. For \( +48 \):
\[
x = \frac{-12 + 48}{8} = \frac{36}{8} = 4.5
\]

2. For \( -48 \):
\[
x = \frac{-12 - 48}{8} = \frac{-60}{8} = -7.5
\]

Since we are looking for the positive solution, we have:

\[
\boxed{4.5}
\]