Answer
$x = -2$
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:
$(\sqrt {-3x - 5})^2 = (x + 3)^2$
$-3x - 5 = (x + 3)^2$
We turn our attention to the right side of the equation to expand the binomials:
$-3x - 5 = (x + 3)(x + 3)$
Use the FOIL method to expand the right side of the equation. With the FOIL method, we multiply the first terms first, then the outer terms, then the inner terms, and, finally, the last terms:
$-3x - 5 = (x)(x) + 3x + 3x + (3)(3)$
$-3x - 5 = x^2 + 3x + 3x + 9$
$-3x - 5 = x^2 + 6x + 9$
Rewrite the equation so that all terms are on one side and $0$ is on the other side::
$0=x^2 + 6x + 9 +3x+5$
$0=x^2 + 9x + 14$
$x^2+9x+14=0$
We solve by factoring. To factor a quadratic equation in the form $ax^2 + bx + c = 0$, we look at factors of the product of $a$ and $c$ such that, when added together, equal $b$.
For the equation $x^2 + 9x + 14 = 0$, $(a)(c)$ is $(7)(2)$, or $14$, but when added together will equal $b$ or $9$. Both factors will be positive.
We came up with the following possibilities:
$(a)(c)$ = $(14)(1)$
$b = 15$
$(a)(c)$ = $(7)(2)$
$b = 9$
The second pair works. Let's take a look at our factorization:
$(x + 7)(x + 2) = 0$
The Zero-Product Property states that if the product of two factors equals zero, then either one of the factors is zero or both factors equal zero. We can, therefore, set each factor to $0$, then solve each equation:
First factor:
$x + 7 = 0$
$x = -7$
Second factor:
$x + 2 = 0$
$x = -2$
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 $x = -7$ first:
$(-3(-7) - 5)^{1/2} = -7 + 3$
Evaluate parentheses first, according to order of operations:
$(21 - 5)^{1/2} = -7 + 3$
Subtract what is within the parentheses:
$16^{1/2} = -7 + 3$
Take the square root of $16$:
$4 = -7 + 3$
Now add the right side of the equation:
$4 = -4$
The left and right sides are not equal; therefore, this solution is extraneous.
Let's check $x = -2$:
$(-3(-2) - 5)^{1/2} = -2 + 3$
Evaluate parentheses first, according to order of operations:
$(6 - 5)^{1/2} = -2 + 3$
Subtract within the brackets first:
$1^{1/2} = -2 + 3$
Take the square root of $1$:
$1 = -2 + 3$
Now add the right side of the equation:
$1 = 1$
The left and right sides are equal; therefore, this is the correct solution.
The solution is $x = -2$.
