Step-by-step solution of algebraic equations involving square roots and quadratic expressions.

Let’s solve problem #18 step by step.

We are given:

√(2x + 30) = x + 3

Step 1: Square both sides to get rid of the square root.

Left side squared: (√(2x + 30))² = 2x + 30
Right side squared: (x + 3)² = x² + 6x + 9 ← Remember: (a+b)² = a² + 2ab + b²

So now we have:

2x + 30 = x² + 6x + 9

Step 2: Move all terms to one side to make a quadratic equation.

Subtract 2x and 30 from both sides:

0 = x² + 6x + 9 - 2x - 30
Simplify:

x² + 4x - 21 = 0

Step 3: Factor the quadratic.

We need two numbers that multiply to -21 and add to +4.
Those numbers are +7 and -3.

So:

(x + 7)(x - 3) = 0

That gives us two possible solutions:

x = -7 or x = 3

Step 4: Check for extraneous solutions — because squaring both sides can create fake answers.

First, check x = -7:

Left side: √(2*(-7) + 30) = √(-14 + 30) = √16 = 4
Right side: -7 + 3 = -4
4 ≠ -4 → So x = -7 is NOT valid.

Now check x = 3:

Left side: √(2*3 + 30) = √(6 + 30) = √36 = 6
Right side: 3 + 3 = 6
6 = 6 → This works!

✔ Only x = 3 is a valid solution.

Final Answer:
3