To solve the equation ( 4x^2 + 12x + 9 = (x + 4)^2 ), we first need to simplify the right side of the equation.
We can expand ((x + 4)^2):
[
(x + 4)^2 = x^2 + 8x + 16
]
Now, we can rewrite the equation:
[
4x^2 + 12x + 9 = x^2 + 8x + 16
]
Next, we move all terms to one side of the equation to set it equal to zero:
[
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 + 4x - 7 = 0
]
Now, we can solve this quadratic equation using the quadratic formula:
[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
]
Here, (a = 3), (b = 4), and (c = -7). We will calculate the discriminant:
[
b^2 - 4ac = 4^2 - 4 \cdot 3 \cdot (-7) = 16 + 84 = 100
]
Now we can substitute back into the quadratic formula:
[
x = \frac{-4 \pm \sqrt{100}}{2 \cdot 3}
]
[
x = \frac{-4 \pm 10}{6}
]
This gives us two potential solutions:
(x = \frac{-4 + 10}{6} = \frac{6}{6} = 1)
(x = \frac{-4 - 10}{6} = \frac{-14}{6} = \frac{-7}{3})
Thus, the solutions to the equation (4x^2 + 12x + 9 = (x + 4)^2) are:
[
x = 1 \quad \text{and} \quad x = -\frac{7}{3}
]
