To solve the equation √(x + 10) – 4 = x, we start by isolating the square root. We can do this by adding 4 to both sides:
√(x + 10) = x + 4
Next, we square both sides to eliminate the square root:
x + 10 = (x + 4)²
Expanding the right side gives us:
x + 10 = x² + 8x + 16
We then rearrange the equation to set it to zero:
0 = x² + 8x + 16 – x – 10
This simplifies to:
0 = x² + 7x + 6
Now we can factor this quadratic equation:
0 = (x + 6)(x + 1)
Setting each factor equal to zero gives us the possible solutions:
x + 6 = 0 → x = -6
x + 1 = 0 → x = -1
So, we have two potential solutions: x = -6 and x = -1. However, we need to check these solutions in the original equation to avoid extraneous solutions:
For x = -6:
√(-6 + 10) – 4 = -6
√4 – 4 = -6
2 – 4 = -6 (True)
For x = -1:
√(-1 + 10) – 4 = -1
√9 – 4 = -1
3 – 4 = -1 (True)
Both values satisfy the original equation, so the solutions to the equation √(x + 10) – 4 = x are x = -6 and x = -1.