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Find the solution set of two to the power of 𝑥 squared equals four to the power of six 𝑥 minus 16 in the set of real numbers.
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In order to solve this problem, first of all, we want to have a quick look at the right-hand side of the equation.
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We can see that the right-hand side of the question is four to the power of six 𝑥 minus 16.
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And a key bit of information to help us with this problem is that four is equal to two to the power of two or two squared.
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And the reason this is interesting is because we want to actually get both sides of our equations involving the same base number.
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And in this case, it’s two.
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So it’s great since as we know that four is equal to two to the power of two, what we can do is we can actually substitute this into our equation.
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So therefore, our equation now becomes: two to the power of 𝑥 squared is equal to two to the power of — and then we have two, because it’s two to the power of two that gives us our four, and then that’s multiplied by six 𝑥 minus 16.
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So great, we’ve actually achieved what we were setting out to do.
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And that’s actually to get the same base on both sides of our equations.
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And as we have the same base on each side of our equation, what we can actually do now is we can actually equate the exponents.
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So we get that 𝑥 squared is equal to two multiplied by six 𝑥 minus 16.
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Great, so we’ve now got an equation.
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And let’s solve to find 𝑥.
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So the first thing we’ll do to solve the equation is actually expand the parentheses.
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So I start with two multiplied by six 𝑥 which gives me 12𝑥.
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And then I’m gonna have two multiplied by negative 16 which gives me negative 32.
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So great, my equation will now be 𝑥 squared is equal to 12𝑥 minus 32.
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As we’re actually looking to solve this equation and find 𝑥, what I’m now gonna do is I’m actually gonna subtract 12𝑥 and add 32 to each side.
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So what we actually have, our equation equal to zero.
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So now we have a quadratic equation which is equal to zero.
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We have 𝑥 squared minus 12𝑥 plus 32 is equal to zero.
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And on inspection of this quadratic, we can see that we can factor it to solve it.
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So we get 𝑥 minus four multiplied by 𝑥 minus eight is equal to zero.
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So then we got this because the sum of negative four and negative eight is negative 12 which is our coefficient of 𝑥.
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And the product of negative four and negative eight is equal to positive 32.
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So, great.
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Okay, they’re our factors.
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And now, let’s find 𝑥.
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So now as we want to find 𝑥, in order to do that, what we need to do is set both our parentheses equal to zero.
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Because in order for the equation to equal zero, then one of our parentheses will also have to equal zero.
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So first, well we’re gonna start with 𝑥 minus four is equal to zero.
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So add four to each side of the equation.
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We’re gonna get 𝑥 is equal to four.
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So great, that’s our first solution.
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Then we move across to 𝑥 minus eight is equal to zero.
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So we just add eight to each side of the equation.
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And there we have 𝑥 is equal to eight.
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Great, so that’s the other solution of our equation.
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So therefore, we can say that the solution set of two to the power of 𝑥 squared is equal to four to the power of six 𝑥 minus 16 in the set of real numbers is: eight, four.
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And just a quick reminder, the key part about a question like this is to make sure that we get both sides of the equation in the same base.
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And in order to do that, look out for a question where you have, say, a two and a four or a two and an eight, a two and a 16 or a three and nine, a three and a 27.
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So keep an eye, as that’s the kind of thing that you’ll see in a question like this.
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And that’s how you’d start the question and move on to solve and find the solution set.