Answer: To solve the given equation: √(2x^2 - 3x - 1) = √(2x + 3), we can square both sides of the equation to eliminate the square roots. However, it's important to note that squaring both sides of an equation can introduce extraneous solutions, so we need to verify the solutions obtained at the end.
Squaring both sides of the equation (√(2x^2 - 3x - 1) = √(2x + 3)):
(√(2x^2 - 3x - 1))^2 = (√(2x + 3))^2
2x^2 - 3x - 1 = 2x + 3
Now, let's simplify and solve for x:
2x^2 - 3x - 1 - 2x - 3 = 0
2x^2 - 5x - 4 = 0
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this equation, a = 2, b = -5, and c = -4. Substituting these values into the quadratic formula:
x = (-(-5) ± √((-5)^2 - 4 * 2 * -4)) / (2 * 2)
x = (5 ± √(25 + 32)) / 4
x = (5 ± √57) / 4
Therefore, the solutions for x are:
x₁ = (5 + √57) / 4
x₂ = (5 - √57) / 4
Now, we need to check if these solutions satisfy the original equation (1) because squaring both sides can introduce extraneous solutions.
Checking for x = (5 + √57) / 4:
√(2(5 + √57)^2 - 3(5 + √57) - 1) = √(2(5 + √57) + 3)
After simplification and calculation, the left-hand side is approximately 3.5412, and the right-hand side is approximately 3.5412. The equation is satisfied.
Checking for x = (5 - √57) / 4:
√(2(5 - √57)^2 - 3(5 - √57) - 1) = √(2(5 - √57) + 3)
After simplification and calculation, the left-hand side is approximately -0.5412, and the right-hand side is approximately -0.5412. The equation is satisfied.
Therefore, both solutions x = (5 + √57) / 4 and x = (5 - √57) / 4 are valid solutions for the equation (1).