Exercise 30 Page 395

To solve equations with a variable expression raised to a rational exponent, we first want to make sure the variable expression is isolated. Then we raise each side of the equation to the reciprocal of the rational exponent. Remember, if $m$ is even, then ${\left(x^{\frac{m}{n}}\right)}^{\frac{n}{m}}=|x|.$ In this case $m=1,$ so we do not need to worry about this.

Finding the Solutions

We will first isolate the variable expression raised to a rational exponent, and then raise each side of the equation to the power of $2.$ We now have a quadratic equation, and we need to find its roots. To do it, let's identify the values of $a,$ $b,$ and $c.$ We can see that $a=1,$ $b=\text{-}5,$ and $c=4.$ Let's substitute these values into the Quadratic Formula. Using the Quadratic Formula, we found that the solutions of the given equation are $x=\frac{5\pm 3}{2}.$ Therefore, the solutions are $x_1=1$ and $x_2=4.$ Let's check them to see if we have any extraneous solutions.

Checking the Solutions

We will check $x_1=1$ and $x_2=4$ one at a time. Let's start by substituting $x=1$ into the original equation. We got a true statement, so $x=1$ is a solution. Let's now substitute $x=4.$ In this case we also got a true statement. Therefore, $x=1$ and $x=4$ are the solutions of the original equation.