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