### Video Transcript

Determine the type of the roots of the equation π₯ minus nine minus π₯ multiplied by π₯ minus five equals zero.

Well, the first thing that we want to do in our problem is, in fact, distribute across our parentheses and then simplify. So weβre gonna do that. Weβre gonna multiply the negative π₯ in front of the parentheses by the π₯ and the negative five within the parentheses. And when we do that, what weβre gonna get is π₯ minus nine minus π₯ squared plus five π₯ equals zero. And then what we want to do is, in fact, rearrange it to be in this form. And this form is the form of our quadratic, ππ₯ squared plus ππ₯ plus π equals zero.

So when we do that, what weβre gonna get is negative π₯ squared. And then weβve got plus six π₯ because weβve got positive five π₯ and weβve got π₯. So we add them together; it gives us positive six π₯ and then minus nine. And this is equal to zero.

Well, now because we want to try and find the type of the roots of the equation, what weβre gonna use is something called the discriminant. And the discriminant is found by squaring π and then subtracting four multiplied by ππ. But what are π, π, and π? Well, if we have a look above when we have our quadratic in the form ππ₯ squared plus ππ₯ plus π, weβve got π is the coefficient of our π₯ squared term, π is the coefficient of our π₯-term, and π is the numerical value at the end.

But we might ask the question, well, why is the discriminant actually useful? Well, itβs useful because what it does is it tells us the type of the roots of our quadratic. Because if π squared minus four ππ is greater than zero, then the roots are real and different. If π squared minus four ππ is equal to zero, then the roots are real and the same. And if itβs π squared minus four ππ is less than zero, then the roots are in fact complex and not real.

So if we take a look at our quadratic, what we have are our π, π, and π. π is negative one, π is six, and π is negative nine. Well, then, if we look at our discriminant, weβre gonna get six squared, cause thatβs our π squared, minus four multiplied by negative one multiplied by negative nine. Well, this is gonna give us 36, because six squared is 36, minus 36. And thatβs because if we have four multiplied by negative one, thatβs negative four multiplied by a negative nine. Well, a negative multiplied by negative is a positive, which gives us 36. So weβve got 36 minus 36. Well, this is gonna be equal to zero.

So therefore, we can say that our discriminant or π squared minus four ππ is gonna be equal to zero. So therefore, we can say an answer to the question, what is the type of the roots of the equation π₯ minus nine minus π₯ multiplied by π₯ minus five equals zero, well, they are real and equal. And thatβs because the discriminant is equal to zero.