Solve the Quadratic Equation: Finding X in 7x + 1 + (2x + 3)^2 = (4x + 2)^2

To solve the equation $7x + 1 + (2x + 3)^2 = (4x + 2)^2$, we follow these steps:

• Step 1: Expand both sides using the square of a binomial formula.
• Step 2: Simplify the equation to form a standard quadratic equation.
• Step 3: Use the quadratic formula to find the roots of the equation.

Step 1: Expand the squares.

The left side: $(2x + 3)^2 = 4x^2 + 12x + 9$.

The right side: $(4x + 2)^2 = 16x^2 + 16x + 4$.

Step 2: Substitute back into the original equation and simplify:

$7x + 1 + 4x^2 + 12x + 9 = 16x^2 + 16x + 4$.

Combine like terms:

$4x^2 + 19x + 10 = 16x^2 + 16x + 4$.

Step 3: Move all terms to one side:

$4x^2 + 19x + 10 - 16x^2 - 16x - 4 = 0$.

Which simplifies to:

$-12x^2 + 3x + 6 = 0$.

Step 4: Divide by -3 to simplify:

$4x^2 - x - 2 = 0$.

Step 5: Use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 4$, $b = -1$, $c = -2$.

Calculate the discriminant:

$b^2 - 4ac = (-1)^2 - 4 \cdot 4 \cdot (-2) = 1 + 32 = 33$.

Calculate the roots:

$x = \frac{1 \pm \sqrt{33}}{8}$.

Therefore, the solution to the problem is $x = \frac{1 \pm \sqrt{33}}{8}$.