### Video Transcript

Determine the type of the roots of the equation two π₯ minus four all squared plus 17 equals zero.

So to help us solve this problem, what Iβve first done is rewritten it. So weβve got two π₯ minus four multiplied by two π₯ minus four, cause we had two π₯ minus four squared plus 17 equals zero. So we can actually see here that what we need to do first is actually expand the parentheses.

As weβre gonna first of all expand the parentheses, what we do first is weβre gonna multiply the first term in each of our parentheses. So that is two π₯ multiplied by two π₯, which gives us four π₯ squared. Then Iβm gonna multiply the first term in our first parentheses by the second term in our second parentheses. And thatβs gonna be two multiplied by negative four, which gives us negative eight π₯.

Remember to take care with the negatives and positives. And then we have another negative eight π₯ because we have the second term in the first parentheses multiplied by the first term in the second parentheses. So we have negative four multiplied by two π₯, which gives us negative eight π₯. And then finally, we multiply the last two terms. So we have negative four multiplied by negative four, which gives us positive 16.

Brilliant! And then finally weβve still got the plus 17 on the end. And then thatβs all equal to zero. Okay, great! So now letβs simplify. So if I simplify, I get four π₯ squared. And then Iβve got negative eight π₯ minus eight π₯, which gives me negative 16π₯. So we now got four π₯ squared minus 16π₯. And now we have 16 plus 17, so we get plus 33. And this is all equal to zero.

Okay, great! So weβve now expanded and weβve simplified. But why have we done this? Why do we want it in this form? Well, we put it in this form because we actually we want these. We want π, we want π, and we want π. We want these because actually we want to use the discriminant. And the discriminant is π squared minus four ππ. And why do we want this? Well, we want this because the discriminant is very useful because actually enables us to tell what type of roots an equation has.

So first of all, if we have π squared minus four ππ is greater than zero so therefore is positive, that tells us that our equation is gonna have real and different roots. However, if we have π squared minus four ππ being equal to zero, then this tells us that our equation has real and repeated roots. Or the roots are real and the same.

So we have π squared minus four ππ is less than zero. And if this is the case, then we know that our roots are complex and not real. Okay, great! So we know what the discriminant is. We now know how to use it and what it means. Letβs try and use it to determine the type of the roots of our equation. So therefore weβre gonna use the discriminant.

Weβve got π squared minus four ππ. What we do now is weβre gonna substitute our values for π, π, and π in. So weβre gonna get negative 16 all squared minus four multiplied by four multiplied by 33, which is gonna be equal to 256 minus 528, which gives us a final answer for our discriminant of negative 272.

Okay, great! So we now we got this value. We can actually look back at our description for the discriminant to see what this actually means. Well, if we have a look across, we can actually see that itβs gonna be the third situation, actually gonna be the one that weβre using, because π squared minus four ππ is actually less than zero because weβve got a negative answer of negative 272.

So therefore we can say that the roots of our equation are complex and not real. And this is because π squared minus four ππ is equal to negative 272, which is less than zero. Okay, great! So weβve got to our final answer.

Just do a quick recap of what weβve done. So first of all, make sure that you get your equation into the form ππ₯ squared plus ππ₯ plus π, which we did here. And then once youβve got that, substitute your values for π, π, and π into the discriminant, which is π squared minus four ππ, to get your value, which we did. And we got negative 272.

And then finally, once youβve got your value for your discriminant, you have to decide, is it greater than, less than, or equal to zero? And this will tell you what roots your equation has.