Question Video: Finding the Value of an Unknown in a Quadratic Equation given One of Its Roots | Nagwa Question Video: Finding the Value of an Unknown in a Quadratic Equation given One of Its Roots | Nagwa

# Question Video: Finding the Value of an Unknown in a Quadratic Equation given One of Its Roots Mathematics • First Year of Secondary School

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Given that β1 is one of the roots of the equation π₯Β² + ππ₯ + 2 = 0, find the value of π and the value of the other root.

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### Video Transcript

Given that negative one is one of the roots of the equation π₯ squared plus ππ₯ plus two equals zero, find the value of π and the value of the other root.

So what we have here is a quadratic equation π₯ squared plus ππ₯ plus two equals zero. Well, the key bit of information from this problem is that weβre told that negative one is one of the roots of the equation. So therefore, what we know is the factored form of the equation must be π₯ plus one multiplied by π₯ plus another number, which weβre gonna call π, is equal to zero. And we know that the first parentheses is π₯ plus one because weβre told that negative one is one of the roots. So therefore, that means that the π₯-value of negative one must make one of the parentheses equal to zero and negative one add one is equal to zero.

So now what weβre gonna do is distribute across our parentheses. So what weβre gonna do is multiply both terms in the first set of parentheses by both terms in the second set of parentheses. So first of all, weβve got π₯ multiplied by π₯, which is π₯ squared. Then we have π₯ multiplied by π, which is ππ₯. Then weβre gonna move on to the positive one. So we have positive one multiplied by π₯, which is just π₯, and then, finally, one multiplied by π, which gives us π. Okay, great. So we now have our quadratic π₯ squared plus ππ₯ plus π₯ plus π. Well, if we take a look at our original quadratic, we can see that the numerical value on the end, so the number without any π₯-term, is positive two. So therefore, this is what π must be equal to because π on its own is the only non-π₯-term that we have.

Well, as weβve worked out that π is equal to two, then this tells us that the other root must be negative two. And thatβs because, for the same reasons we gave before, if we want to make the right parentheses equal to zero, then if π is equal two, then the π₯-value that will make that equals zero is negative two because negative two add two is zero. So now thereβs just one more part of the question left to complete, and that is to find the value of π.

Well, if we substitute in our π-value of two, what weβre gonna get is π₯ squared plus two π₯ plus π₯ plus two equals zero. So now what we can is collect like terms. And when we do that, weβre left with a quadratic π₯ squared plus three π₯ plus two equals zero. Well, as π is the coefficient of π₯ and in our new quadratic the coefficient of π₯ is three, we can therefore say that π is equal to three. So therefore, we found the value of the other root, which is negative two, and the value of π, which is three.

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