Answer
$x = 1$
Work Step by Step
We need to get rid of the fractional exponents to be able to solve this equation; therefore, we will multiply by the reciprocal of the fractional exponent $1/2$, which is $2$. We will now square both sides of the equation:
$((5 - x)^{1/2})^2 = (x + 1)^2$
Simplify the equation:
$5 - x = (x + 1)(x + 1)$
$5 - x = (x)(x) +1x + 1x + (1)(1)$
$5 - x = x^2 + 2x + 1$
Rewrite the equation so one side is $0$:
$0=x^2 + 2x + 1 - 5 + x\\
0=x^2+3x-4\\
x^2+3x-4=0$
Factor the trinomial:
$(x-1)(x+3)=0$
Use the Zero-Product Property by equating each factor to $0$, then solve each equation.
First factor:
$x - 1 = 0$
$x = 1$
Second factor:
$x + 4 = 0$
$x = -4$
To check if our solutions are correct, we plug our solutions back into the original equation to see if the left and right sides equal one another.
Let's plug in $x = 1$ first:
$(5 - 1)^{1/2} = 1 + 1$
Evaluate parentheses first, according to order of operations:
$(4)^{1/2} = 1 + 1$
Let's evaluate the exponent next:
$2 = 1 + 1$
Now add the right side of the equation:
$2 = 2$
The left and right sides are equal; therefore, this solution is correct.
Let's check $x = -4$:
$(5 - (-4))^{1/2} = -4 + 1$
Evaluate parentheses first, according to order of operations:
$(9)^{1/2} = -4 + 1$
Let's evaluate the exponent next:
$3 = -4 + 1$
Now add the right side of the equation:
$3 = -3$
The left and right sides are not equal; therefore, this is an extraneous solution.
The solution is $x = 1$.
