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