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Question

ScanSolve AI Tutor · Accepted Answer

The image contains a math problem asking to solve for x in the equation:
√(2x + 3) - √(x - 2) = 2

Isolate one of the square roots.
Add √(x - 2) to both sides of the equation:
√(2x + 3) = 2 + √(x - 2)
Square both sides of the equation to eliminate the square root on the left.
(√(2x + 3))² = (2 + √(x - 2))²
2x + 3 = 2² + 2(2)√(x - 2) + (√(x - 2))²
2x + 3 = 4 + 4√(x - 2) + (x - 2)
Simplify the equation and isolate the remaining square root term.
2x + 3 = 4 + 4√(x - 2) + x - 2
2x + 3 = x + 2 + 4√(x - 2)
Subtract x and 2 from both sides:
2x - x + 3 - 2 = 4√(x - 2)
x + 1 = 4√(x - 2)
Square both sides again to eliminate the remaining square root.
(x + 1)² = (4√(x - 2))²
x² + 2x + 1 = 16(x - 2)
x² + 2x + 1 = 16x - 32
Rearrange the equation into a quadratic form (ax² + bx + c = 0).
x² + 2x + 1 - 16x + 32 = 0
x² - 14x + 33 = 0
Solve the quadratic equation.
This can be done by factoring, completing the square, or using the quadratic formula. Let's try factoring. We need two numbers that multiply to 33 and add to -14. These numbers are -3 and -11.
(x - 3)(x - 11) = 0
Find the possible values for x.
x - 3 = 0 ⟹ x = 3
x - 11 = 0 ⟹ x = 11
Check for extraneous solutions by substituting the values back into the original equation.
For x = 3:
√(2(3) + 3) - √(3 - 2) = √(6 + 3) - √(1) = √9 - 1 = 3 - 1 = 2. This is correct.

For x = 11:
√(2(11) + 3) - √(11 - 2) = √(22 + 3) - √(9) = √25 - 3 = 5 - 3 = 2. This is also correct.

Both solutions are valid.

The solutions are x = 3 and x = 11.

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