Problem

Solve the radical equation: \(\sqrt{3x-2} = 2x + 1\).

Answer

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Answer

Step 3: Solve the quadratic equation. Since the discriminant \(b^2 - 4ac = 1^2 - 4(4)(3) = -47\) is less than zero, there are no real solutions to this equation.

Steps

Step 1 ：Step 1: Square both sides of the equation to eliminate the square root. \(\left(\sqrt{3x-2}\right)^2 = \left(2x + 1\right)^2\) becomes \(3x - 2 = 4x^2 + 4x + 1\).

Step 2 ：Step 2: Rearrange the equation to form a quadratic equation. \(4x^2 + 4x + 1 - 3x + 2 = 0\) becomes \(4x^2 + x + 3 = 0\).

Step 3 ：Step 3: Solve the quadratic equation. Since the discriminant \(b^2 - 4ac = 1^2 - 4(4)(3) = -47\) is less than zero, there are no real solutions to this equation.

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