Part A) solve the equation and show all necessary steps. Sqrt(2x+10)-1=x

Question

Part B) one of your solutions is extraneous, which means it is not a valid solution to the problem. which solution is extraneous? Explain how you know.

Answer 1

Part A:

Given equation: √(2x+10) - 1 = x

Step 1: Let's isolate the square root term by adding 1 on both sides:
√(2x+10) = x + 1

Step 2: We'll square both sides of the equation to eliminate the square root:
(√(2x+10))^2 = (x + 1)^2
2x + 10 = x^2 + 2x + 1

Step 3: Rearrange the equation to form a quadratic equation:
0 = x^2 + 2x - 2x + 1 - 10
0 = x^2 - 9

Step 4: Factor the quadratic equation:
0 = (x - 3)(x + 3)

Step 5: Set each factor equal to zero:
x - 3 = 0 or x + 3 = 0

For x - 3 = 0, we get:
x = 3

For x + 3 = 0, we get:
x = -3

Therefore, the solutions to the equation are x = 3 and x = -3.

Part B:

To determine which solution is extraneous, we substitute both values back into the original equation and check for validity.

Substituting x = 3:
√(2(3) + 10) - 1 = 3
√(6 + 10) - 1 = 3
√16 - 1 = 3
4 - 1 = 3
3 = 3

Substituting x = -3:
√(2(-3) + 10) - 1 = -3
√(16) - 1 = -3
4 - 1 = -3
3 = -3

Both solutions satisfy the original equation, so neither solution is extraneous.