Video: Finding the Solution Set of Quadratic Equations over the Set of Irrational Numbers

Find the solution set of 3𝑥²/7 = 3, in ℚ′.

01:39

Video Transcript

Find the solution set of three 𝑥 squared over seven equals three, that isn’t in the set of rational numbers.

So looking at this question, we can actually see this last bit of notation. And I’ve just wanted to explain what that actually meant. And as I said when I read the question through, it means it is not in the set of rational numbers. And that’s because our apostrophe, that means not. And the capital 𝑄 means set of rational numbers. So therefore, we know that our solutions are gonna be irrational numbers.

Okay, so now we can actually solve the equation. So we have three 𝑥 squared over seven equals three. So first of all, we multiply each side of the equation by seven. So we get three 𝑥 squared equals 21. And then we actually divide each side of the equation by three. So we’ve now got 𝑥 squared is equal to seven. And as we’re looking to find 𝑥, what we do now is we’re actually gonna take the square root of each side. So then we get 𝑥 is equal to plus or minus root seven. So then we take a look at the root seven part of our answer. Because for it to actually fulfill the set of solutions that we want, it needs to be in a rational number. And yes, root seven is a surd. And it is an irrational number.

So great, we now know we can carry on and get the solution set. So therefore, we can say that the solution set of three 𝑥 squared over seven equals three, that isn’t in the set of rational numbers, is negative root seven, root seven.

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