What is the solution for x in the equation sqrt(2x + 3) = x + 2?

To solve the equation ( \sqrt{2x + 3} = x + 2 ), we begin by squaring both sides to eliminate the square root. This gives us:

[

2x + 3 = (x + 2)^2

]

Expanding the right side:

[

2x + 3 = x^2 + 4x + 4

]

Next, we rearrange the equation to set everything to one side:

[

0 = x^2 + 4x + 4 - 2x - 3

]

Combining like terms results in:

[

0 = x^2 + 2x + 1

]

This can be factored as:

[

0 = (x + 1)^2

]

Setting the factor equal to zero gives ( x + 1 = 0 ), which means:

[

x = -1

]

After finding this potential solution, we must check if it satisfies the original equation, as squaring both sides can introduce extraneous solutions. Substituting ( x = -1 ) back into the original equation:

[

\sqrt{2(-1) + 3