sqrt of x-1=x-3
square both sides. Gather terms, factor or use the quadratic equation. I will be happy to critique your work.
To solve the equation √(x-1) = x-3, we need to isolate the variable x. Here's how you can do it:
Step 1: Square both sides of the equation to eliminate the square root.
(√(x-1))^2 = (x-3)^2
(x-1) = (x-3)^2
Step 2: Simplify the equation by expanding the right side using the FOIL method.
x - 1 = (x-3)(x-3)
x - 1 = (x^2 - 3x - 3x + 9)
x - 1 = (x^2 - 6x + 9)
Step 3: Rearrange the equation to get all the terms on one side.
x - 1 - x^2 + 6x - 9 = 0
-x^2 + 7x - 10 = 0
Step 4: Now, we can either factor the quadratic equation or use the quadratic formula. Let's factor it:
(x - 2)(-x + 5) = 0
Step 5: Set each factor equal to zero and solve for x.
(x - 2) = 0 or (-x + 5) = 0
For (x - 2) = 0:
x - 2 = 0
x = 2
For (-x + 5) = 0:
-x + 5 = 0
x = 5
So, the equation √(x-1) = x-3 has two solutions: x = 2 and x = 5.