Final answer:
To solve the radical equation √(x+10) = x-2, you square both sides to eliminate the square root, rearrange the resulting quadratic equation, factor it, and then check the potential solutions in the original equation. The valid solution is x = 6.
Step-by-step explanation:
To solve the radical equation √(x+10) = x-2, we need to isolate the square root on one side of the equation and then square both sides to eliminate the radical. Here's how it's done step by step:
- Start with the equation √(x+10) = x-2.
- Square both sides of the equation to get rid of the square root: (√(x+10))^2 = (x-2)^2, which simplifies to x+10 = x^2 - 4x + 4.
- Rearrange the terms to set the equation to zero: x^2 - 4x + 4 - x - 10 = 0, simplifying to x^2 - 5x - 6 = 0.
- As this is a quadratic equation, we can factor it: (x - 6)(x + 1) = 0.
- Thus, there are two possible solutions for x: x = 6 or x = -1.
- However, we must check these solutions in the original equation to ensure they are valid, because squaring both sides of an equation can introduce extraneous solutions.
- Checking x = 6 in the original equation: √(6+10) = 6-2, √(16) = 4, which is true because 4 = 4.
- Checking x = -1 in the original equation: √(-1+10) = -1-2, √(9) = -3, which is false because 3 ≠ -3.
- Therefore, the only solution is x = 6.