What is the solution set for the equation sqrt(3x + 7) = x?

To solve the equation ( \sqrt{3x + 7} = x ), we start by squaring both sides to eliminate the square root. This gives us:

[

3x + 7 = x^2

]

Next, we rearrange the equation to set it to zero:

[

x^2 - 3x - 7 = 0

]

Now, we can apply the quadratic formula, which is defined as ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ), where ( a = 1 ), ( b = -3 ), and ( c = -7 ).

Calculating the discriminant:

[

b^2 - 4ac = (-3)^2 - 4(1)(-7) = 9 + 28 = 37

]

Now we can plug this back into the quadratic formula:

[

x = \frac{-(-3) \pm \sqrt{37}}{2 \cdot 1} = \frac{3 \pm \sqrt{37}}{2}

]

This results in two potential solutions:

[

x = \frac{