4x2+12x+9=(x-4)2

Let's solve the equation step by step. The equation you provided is:

\[ 4x^2 + 12x + 9 = (x - 4)^2 \]

First, let's expand the right side:

\[ (x - 4)^2 = x^2 - 8x + 16 \]

Now, substitute this back into the equation:

\[ 4x^2 + 12x + 9 = x^2 - 8x + 16 \]

Now, move all terms to one side of the equation to set it to 0:

\[ 4x^2 + 12x + 9 - x^2 + 8x - 16 = 0 \]

Combine like terms:

\[ (4x^2 - x^2) + (12x + 8x) + (9 - 16) = 0 \]

This simplifies to:

\[ 3x^2 + 20x - 7 = 0 \]

Now, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 3 \), \( b = 20 \), and \( c = -7 \).

Calculating the discriminant (\( b^2 - 4ac \)):

\[ b^2 - 4ac = 20^2 - 4 \cdot 3 \cdot (-7) = 400 + 84 = 484 \]

Now, applying the quadratic formula:

\[ x = \frac{-20 \pm \sqrt{484}}{2 \cdot 3} \]

Calculating \( \sqrt{484} = 22 \):

\[ x = \frac{-20 \pm 22}{6} \]

Now, we find the two solutions:

1. \( x = \frac{-20 + 22}{6} = \frac{2}{6} = \frac{1}{3} \)

2. \( x = \frac{-20 - 22}{6} = \frac{-42}{6} = -7 \)

Thus, the solutions to the equation \( 4x^2 + 12x + 9 = (x-4)^2 \) are:

\[ x = \frac{1}{3} \quad \text{and} \quad x = -7 \]