Video Transcript
Find the solution set of four to the power of 𝑥 squared minus two 𝑥 is equal to 0.25 in the set of real numbers.
In this question, we’re given an equation involving the variable 𝑥. And we need to find the entire solution set to this equation in the set of real numbers. So we need to find all values of 𝑥 which make this equation true. To do this, let’s start by taking a look at the equation we’re given. We can see that this equation involves a power function. We have four raised to the power of some function in 𝑥. And it’s difficult to solve equations like this directly. It would be easier if this was also equal to four raised to some power because then we could just equate the exponents.
And in fact, in this case, it’s possible to do this. First we need to notice that 0.25 is actually equal to one divided by four. And then we can rewrite one-quarter by using our laws of exponents. We know one over 𝑎 is actually equal to 𝑎 raised to the power of negative one. So in fact, instead of writing 0.25 as one-quarter, we can instead write it as four to the power of negative one.
So now we need to solve the equation four to the power of 𝑥 squared minus two 𝑥 is equal to four to the power of negative one. And for these two to be equal, their exponents have to be equal. Therefore, we must have that 𝑥 squared minus two 𝑥 is equal to negative one. And we know how to solve this for 𝑥. We’ll start by adding one to both sides of our equation. This gives us that 𝑥 squared minus two 𝑥 plus one is equal to zero. And now we can see this is a quadratic equation in 𝑥. And there’s a few different ways of solving this. For example, we could use the quadratic formula. However, this isn’t necessary.
We could also notice that negative one multiplied by negative one is equal to positive one and negative one plus negative one is equal to negative two. So we can factor our quadratic fully. We get 𝑥 minus one multiplied by 𝑥 minus one should be equal to zero. And now, finally, if the product of two numbers is equal to zero, this means one of our numbers has to be equal to zero. So one of our factors must be equal to zero. So we just solve each factor individually is equal to zero. And we see in both cases we just get that 𝑥 is equal to one.
And we might want to leave our answer just as 𝑥 is equal to one. However, the question is specifically asking for the solution set. So we should give our answer in terms of set notation. It’s just the set containing the number one. Therefore, what we were able to show the solution set of the equation four to the power of 𝑥 squared minus two 𝑥 is equal to 0.5 in the set of real numbers is just the set containing the number one.
