= √x + 1 = x - 5
= x + 1 = x² - 10x + 25
= x + 1 - x² + 10x - 25 = 0
= 11x + 1 - x² - 25 = 0
= 11x - 24 - x² = 0
= -x² + 11x - 24 = 0
= x² - 11x + 24 = 0
= x² - 3x - 8x + 24 = 0
= x × ( x - 3 ) - 8x + 24 = 0
= x × ( x - 3 ) - 8 × ( x - 3 ) = 0
= ( x - 3 ) × ( x - 8 ) = 0
x - 3 = 0
x - 8 = 0
x = 3
x = 8
= √3 + 1 = 3 - 5
= √8 + 1 = 8 - 5
√3 + 1 = 3 - 5
= 2 = -2
√8 + 1 = 8 - 5
= 3 = 3
2 = -2
= x ≠ 3
3 = 3
= x = 8
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Answers & Comments
x = 8
Simplify the equation and square both sides of the equation
= √x + 1 = x - 5
= x + 1 = x² - 10x + 25
Move the expression to the left hand side and change its sign
= x + 1 = x² - 10x + 25
= x + 1 - x² + 10x - 25 = 0
Collect like terms and calculate
= x + 1 - x² + 10x - 25 = 0
= 11x + 1 - x² - 25 = 0
= 11x - 24 - x² = 0
Reorder the terms by commutative property
= 11x - 24 - x² = 0
= -x² + 11x - 24 = 0
Change the signs of both sides of the equation
= -x² + 11x - 24 = 0
= x² - 11x + 24 = 0
Rewrite -11x as a difference
= x² - 11x + 24 = 0
= x² - 3x - 8x + 24 = 0
Factor out x and -8 from the expression
= x² - 3x - 8x + 24 = 0
= x × ( x - 3 ) - 8x + 24 = 0
= x × ( x - 3 ) - 8 × ( x - 3 ) = 0
Factor out x - 3 from the expression
= x × ( x - 3 ) - 8 × ( x - 3 ) = 0
= ( x - 3 ) × ( x - 8 ) = 0
Separate into possible cases
x - 3 = 0
x - 8 = 0
Solve the equation for x
x = 3
x = 8
Check the solutions
x = 3
= √3 + 1 = 3 - 5
x = 8
= √8 + 1 = 8 - 5
Simplify the expressions
√3 + 1 = 3 - 5
= 2 = -2
√8 + 1 = 8 - 5
= 3 = 3
x = 3 is not a solution, x = 8 is a solution
2 = -2
= x ≠ 3
3 = 3
= x = 8
The equation has one solution
= x = 8