# Question Video: Determining the Type of the Roots of a Quadratic Equation Mathematics • 9th Grade

Determine the type of the roots of the equation (π₯ β 9) β π₯(π₯ β 5) = 0.

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### 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 weβre gonna do that. Weβre 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 weβre 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, ππ₯ squared plus ππ₯ plus π equals zero.

So when we do that, what weβre gonna get is negative π₯ squared. And then weβve got plus six π₯ because weβve got positive five π₯ and weβve 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 weβre gonna use is something called the discriminant. And the discriminant is found by squaring π and then subtracting four multiplied by ππ. But what are π, π, and π? Well, if we have a look above when we have our quadratic in the form ππ₯ squared plus ππ₯ plus π, weβve 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, itβs useful because what it does is it tells us the type of the roots of our quadratic. Because if π squared minus four ππ is greater than zero, then the roots are real and different. If π squared minus four ππ is equal to zero, then the roots are real and the same. And if itβs π squared minus four ππ 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, weβre gonna get six squared, cause thatβs 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 thatβs because if we have four multiplied by negative one, thatβs negative four multiplied by a negative nine. Well, a negative multiplied by negative is a positive, which gives us 36. So weβve 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 ππ 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 thatβs because the discriminant is equal to zero.