Answer
The solution is $x = 1$.
Work Step by Step
First, let's rewrite the equation so that the radical is isolated on the left side of the equation, so let's go ahead and add $1$ to both sides of the equation:
$\sqrt {x + 3} = x + 1$
We don't want to work with radicals, so we need to transform them. To get rid of radicals, we square both sides of the equation:
$(\sqrt {x + 3})^2 = (x + 1)^2$
Squaring the radical will leave just the binomial:
$x + 3 = (x + 1)^2$
We turn our attention to the right side of the equation to expand the binomial:
$x + 3 = (x + 1)(x + 1)$
Let's 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:
$x + 3 = (x)(x) + x + x + (1)(1)$
Multiply to simplify:
$x + 3 = x^2 + 2x + 1$
Rewrite the equation so the term with the greatest degree lies on the left side of the equation:
$x^2 + 2x + 1 = x + 3$
We move all terms to the left side of the equation so that we are left with $0$ on the right side of the equation:
$x^2 + x - 2 = 0$
We solve by factoring.
To factor a quadratic trinomial in the form $ax^2 + bx + c$, we look at factors of $c$ such that, when added together equals $b$.
For the trinomial $x^2 + x - 2$, $c=-2$ so look for factors of $-2$ than when added together will equal $b$ or $1$. One factor must be negative, and the other must be positive, with the positive factor having the greater absolute value. This is because a negative number multiplied with a positive number will equal a negative number; however, when a negative number is added to a positive number, the result can be negative or positive, depending on which factor has the greater absolute value.
We have the following possibilities:
$-2= (2)(-1)$
$2+(-1) = 1$
$-2= (-2)(1)$
$-2+1 = -1$
The first factorization works so the factored form of the trinomial is $(x+2)(x-1)$.
Thus, the equation above is equivalent to:
$$(x+2)(x-1)=0$$
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:
$x + 2 = 0$
$x = -2$
Second factor:
$x - 1 = 0$
$x = 1$
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 $x = -2$ first:
$\sqrt {-2 + 3} = -2 + 1$
Evaluate what is in brackets first, according to order of operations:
$\sqrt {1} = -2 + 1$
Evaluate square root:
$1 = -2 + 1$
Now add:
$1 \ne -1$
The left and right sides are not equal; therefore, this solution is extraneous.
Let's check $x = 1$:
$\sqrt {1 + 3} = 1 + 1$
Evaluate what is in brackets first, according to order of operations:
$\sqrt {4} = 1 + 1$
Evaluate square root:
$2 = 1 + 1$
Now subtract:
$2 = 2$
The left and right sides are equal; therefore, this is a valid solution.
The solution is $x = 1$.
