What is the solution for x in the equation x + 5 = sqrt(5x + 49)?

To find the solution for ( x ) in the equation ( x + 5 = \sqrt{5x + 49} ), we start by squaring both sides to eliminate the square root. This gives us:

[

(x + 5)^2 = 5x + 49

]

Expanding the left side, we have:

[

x^2 + 10x + 25 = 5x + 49

]

Next, we rearrange the equation to bring all terms to one side:

[

x^2 + 10x + 25 - 5x - 49 = 0

]

This simplifies to:

[

x^2 + 5x - 24 = 0

]

Now we can factor this quadratic equation. We look for two numbers that multiply to (-24) and add to (5). The numbers (8) and (-3) work since (8 \cdot (-3) = -24) and (8 + (-3) = 5). Therefore, we can factor the quadratic as:

[

(x + 8)(x - 3) = 0

\