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 \( x \):
-
For \( +48 \): \[ x = \frac{-12 + 48}{8} = \frac{36}{8} = 4.5 \]
-
For \( -48 \): \[ x = \frac{-12 - 48}{8} = \frac{-60}{8} = -7.5 \]
Since we are looking for the positive solution, we have:
\[ \boxed{4.5} \]