Video: SAT Practice Test 1 • Section 3 • Question 12

Video Transcript

If the equation eight 𝑥 minus three 𝑥𝑦 equals 𝑥 squared plus 16 is graphed in the 𝑥𝑦-plane, at which point will the graph touch the 𝑥-axis?

So in this question, we want to find out at which point the graph will touch the 𝑥-axis. Well, where is this going to be? Well, what we do know is that the 𝑦-coordinate is going to be equal to zero. And that’s because anywhere along the 𝑥-axis, 𝑦 is equal to zero. So therefore, we know that the first thing we can do is substitute in 𝑦 is gonna be equal to zero cause as we’ve already said we want to find out where the graph is gonna touch the 𝑥-axis so our 𝑦-values can be equal to zero.

So when we do this, we’re gonna get eight 𝑥 minus three 𝑥 multiplied by zero is equal to 𝑥 squared plus 16. So now if we tidied this up, we’re gonna get eight 𝑥 because three 𝑥 multiplied by zero is just zero is equal to 𝑥 squared plus 16. So then, what we can do is subtract eight 𝑥 from each side of the equation. And we wanna do that because we want it equal to zero.

So when we do that, we get zero is equal to 𝑥 squared minus eight 𝑥 plus 16, which I’ve just flipped and rewritten as 𝑥 squared minus eight 𝑥 plus 16 equals zero just cause I like to have the 𝑥 terms on the left-hand side.

So now what we wanna do is we want to solve this to find out our 𝑥-value. And the best way to do that is to factor. So we’re gonna factor our quadratic expression. So just to remind ourselves what we do when we factor, it means that we want to find two numbers that will multiply together to give us positive 16, but they’ll add together to give us negative eight. So that’s the coefficient of our 𝑥 term.

So before we put in the values that we’re gonna go and try to find, let’s look at what we first have when we factor a quadratic. When we first factor a quadratic, what we do is we have two parentheses and we also have an 𝑥 term at the beginning of each of them. We have an 𝑥 term at the beginning because we have an 𝑥 squared term at the beginning of our quadratic. And 𝑥 multiplied by 𝑥 gives us 𝑥 squared.

So now, we’re gonna take a look at our factors of 16. Well, we’ve got six and one, eight and two, four and four. However, although we know that they need to multiply together to give us positive 16, they need to add together to give us negative eight. So therefore, we know that our factors must both be negative. And that’s because if we want to make a positive, we have to have a negative multiplied by a negative because we couldn’t have two positives. Because if we had two positives, then they would not added together to give us negative eight.

Okay, so now what we need to do is find our pair of factors they’re going to add together to give us negative eight. Because we’ve got the factors here that make positive 16 when multiplied together, but we need to add them together to make negative eight. Well, if we have negative 16 add negative one, we get negative 17. If we have a negative eight add negative two, we get negative 10. But if we have negative four add negative four, we’re gonna get negative eight.

So therefore, we can now put these factors in our parentheses. So we have 𝑥 minus four multiplied by 𝑥 minus four is equal to zero. And also if we look at our question, we can see that these factors look about right because it says at which point will the graph touch the 𝑥-axis. And it says, “at which point.” So there’s only one point.

So therefore, that tells us that we should have a recurring root. So it means that we should have the same factor in each of our parentheses. And we do! 𝑥 minus four, 𝑥 minus four. So now, let’s find the 𝑥-value.

Well, to find the 𝑥-value, that’s the correct solution to our equation. What we need to do is make 𝑥 minus four equal to zero. And that’s because we need to have one of the parentheses on the left-hand side equal to zero. Well as they’re both 𝑥 minus four, we need to find out what 𝑥-value will make that equal to zero.

So now, what we do is we add four to each side of the equation because we want to solve to find 𝑥. So when we do, we get 𝑥 is equal to four. So great, we found our 𝑥-value or our 𝑥-coordinate. And from earlier, we already mentioned that our 𝑦-coordinate was going to be equal to zero.

So therefore, we can say that if the equation eight 𝑥 minus three 𝑥𝑦 equals 𝑥 squared plus 16 is graphed in the 𝑥𝑦-plane, the point at which the graph will touch the 𝑥-axis is four, zero.