Answer
$x = -4$ or $x = -3$
Work Step by Step
First, we want to isolate the radical:
$\sqrt {3x + 13} = x + 5$
Square both sides of the equation to eliminate the radical:
$3x + 13 = (x + 5)^2$
$3x + 13 = x^2 + 10x + 25$
Move all terms to the left side of the equation and combine like terms:
$3x+13-x^2 - 10x - 25 = 0\\
-x^2 - 7x - 12 = 0$
Divide both sides of the equation by $-1$ so that the $x^2$ term is positive:
$x^2 + 7x + 12 = 0$
We have a quadratic equation, which is in the form $ax^2 + bx + c = 0$. We need to find which factors multiplied together will equal $ac$ but when added together will equal $b$.
In this equation, $ac$ is $12$ and $b$ is $7$. The factors $4$ and $3$ will work.
Let's rewrite the equation in factored form:
$(x + 4)(x + 3) = 0$
Sole using the Zero-Product Property bu equating each factor to $0$< then solve each equation.
First factor:
$x + 4 = 0$
$x = -4$
Second factor:
$x + 3 = 0$
$x = -3$
To check if we have any extraneous solutions, we substitute each solution into the original equation to see if the two sides equal one another.
Let's check the first solution:
$\sqrt {3(-4) + 13} - 5 = -4$
Multiply to simplify:
$\sqrt {-12 + 13} - 5 = -4$
Combine the terms inside the radicand:
$\sqrt {1} - 5 = -4$
Take the square root:
$1 - 5 = -4$
Combine like terms:
$-4 = -4$
The sides are equal; therefore, this is a valid solution.
Let's try the solution $x = -3$:
$\sqrt {3(-3) + 13} - 5 = -3$
Multiply to simplify:
$\sqrt {-9+ 13} - 5 = -3$
Combine the terms inside the radicand:
$\sqrt {4} - 5 = -3$
Take the square root:
$2 - 5 = -3$
Combine like terms:
$-3= -3$
Both sides are equal; therefore, this solution is also valid.
