Question Video: Finding the Solution Set to a Multistep Linear Equation That Includes a Radical Mathematics

Find, in ℝ, the solution set of the equation √5(√3𝑥 − 2) = 4√5.

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Video Transcript

Find, in the set of real numbers, the solution set of the equation root five multiplied by root three 𝑥 minus two is equal to four root five.

There are a couple of approaches we could use to answer this question. For example, we could begin by distributing the parentheses on the left-hand side. However, as the unknown variable 𝑥 appears inside the parentheses, it will be simpler to divide both sides of the equation by root five first. When we do this, we have root five multiplied by root three 𝑥 minus two all divided by root five is equal to four root five over root five. On both sides of the equation, the root fives cancel. And as such, our equation becomes root three 𝑥 minus two is equal to four. We can then add two to both sides of this equation to isolate the 𝑥-term. As four plus two is equal to six, we have root three 𝑥 equals six.

Next, we divide through by the coefficient of 𝑥, in this case root three. And 𝑥 is therefore equal to six over root three. Since the denominator of our fraction is a radical, we need to rationalize the denominator by multiplying the numerator and denominator by root three. Recalling that root three multiplied by root three is three, we have 𝑥 is equal to six root three over three, which in turn simplifies to 𝑥 is equal to two root three. The solution set of the equation root five multiplied by root three 𝑥 minus two is equal to four root five is the single value two root three. We could check this answer by substituting our value of 𝑥 back in to the original equation.