Final answer:
To solve the equation 3x + 2 = 2√(2x + 4√2), we can eliminate the square root by squaring both sides. By expanding and simplifying the equation, we can then solve it as a quadratic equation using the quadratic formula.
Step-by-step explanation:
The given equation is 3x + 2 = 2√(2x + 4√2). To solve for x, we need to isolate x on one side of the equation. Let's start by squaring both sides of the equation to eliminate the square root:
(3x + 2)² = (2√(2x + 4√2))²
Expanding the left side, we get:
9x² + 12x + 4 = 8(2x + 4√2)
Now, distribute the 8 on the right side:
9x² + 12x + 4 = 16x + 32√2
Combine like terms and isolate the radical term:
9x² - 4x - 16x - 32√2 + 4 = 0
9x² - 20x - 32√2 + 4 = 0
This is a quadratic equation. We can solve it by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-(-20) ± √((-20)² - 4(9)(-32√2 + 4))) / (2(9))
x = (20 ± √(400 + 1152√2 - 144)) / 18
Simplifying under the square root:
x = (20 ± √(400 + 1152√2 - 144)) / 18
x = (20 ± √256 + 1152√2) / 18
x = (20 ± (16 + 1152√2)) / 18
The answer is x = (20 + (16 + 1152√2)) / 18 or x = (20 - (16 + 1152√2)) / 18.