Video Transcript
If 𝐿 and 𝑀 are the roots of the equation 𝑥 squared plus 10𝑥 plus nine equals zero, what is the value of 𝐿 squared plus 𝑀 squared?
Well, to solve this problem, what we want to do is find 𝐿 and 𝑀. So we want to find the roots of the equation. Well, there are a number of ways we could do this. We could factor, we could use a quadratic formula, we could complete the square, and there are other methods we could use as well. For this question, what we’re gonna do is we can use the first method I mentioned. And that method is to factor because taking a look at our equation, I can see that we are going to be able to factor what we have here.
So when we’re looking to factor a quadratic, then what we’re doing is we’re looking for a couple of values. And these two values have to do two things. They have to sum to give us our coefficient of 𝑥. So in this case, that’s positive 10 because we must remember the sign. And then their product must give us the numerical value in the end, so in this case, positive nine.
And when we’re factoring a quadratic, what we have is two pairs of parentheses. And within these, the first value is going to be 𝑥. And that’s because the first term in our quadratic is 𝑥 squared. So therefore, if we had 𝑥 multiplied by 𝑥, this would give us our 𝑥 squared. Then the terms inside the parentheses after this are gonna depend on the sum and product of the numbers that we just mentioned.
Well, in this problem, we can see that the two numbers are gonna be positive nine and positive one. And that’s because nine multiplied by one is nine. And because it’s positive, they both have to be positive or both be negative. But we know that both are gonna be negative because if we add nine and one, we get positive 10.
Well, we’ve factored our quadratic. But does this mean we’ve solved it? Well, no, cause what we need to do now is find the roots. Well, the roots or solutions to the equation are going to be negative nine or negative one. And we know that the roots are gonna be negative nine or negative one because what they need to be is the value of 𝑥 that will make each of our parentheses equal to zero. And that’s because on the right-hand side, we have zero. So that means that one of our parentheses must be equal to zero. So negative nine plus nine is equal to zero, and negative one plus one is equal to zero.
So therefore, what we’re gonna do is we’re gonna call 𝐿 negative nine and 𝑀 negative one because these are the roots to the equation. Now, what we want to do is find the value of 𝐿 squared plus 𝑀 squared. And because we’re trying to find a value of 𝐿 squared plus 𝑀 squared, it wouldn’t matter which way round we called 𝐿 and 𝑀. So what we’re gonna get is that 𝐿 squared plus 𝑀 squared is gonna be equal to negative nine squared plus negative one squared, which is going to be equal to 81 plus one. And that’s because a negative multiplied by a negative is a positive. So we’ve got 81 plus one.
So therefore, we can say that if 𝐿 and 𝑀 are the roots of the equation 𝑥 squared plus 10𝑥 plus nine equals zero, then the value of 𝐿 squared plus 𝑀 squared is 82. So we got to our final answer.
But what we are gonna do is just double-check this. And we’re gonna double-heck this by using another method to solve our equation. And this other method is called the quadratic formula. Well, the method we’re gonna use to check is the quadratic formula. And what the quadratic formula is is a way of solving quadratic equations.
So we’ve got 𝑥 is equal to negative 𝑏 plus or minus the square root of 𝑏 squared minus four 𝑎𝑐 over two 𝑎. And this is when we have a quadratic in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. Well, we can see that from our quadratic equation, our 𝑎 is gonna be equal to one cause the coefficient of 𝑥 squared is one, 𝑏 10, and 𝑐 nine. So for our checking method, what we’re gonna do is plug our values into the quadratic formula. And when we do this, we’re gonna get 𝑥 is equal to negative 10 plus or minus square root of 10 squared minus four multiplied by one multiplied by nine over two multiplied by one.
So we’re gonna say that 𝑥 is equal to negative 10 plus or minus the square root of 100 minus 36 over two. So this is gonna give 𝑥 is equal to negative 10 plus eight over two or negative 10 minus eight over two, which would give us our two solutions as 𝑥 equals negative one or negative nine, which is what we got with our first method. So this is definitely correct. We don’t need to do the rest of the working out because this would’ve been the same. So we know that the correct answer is definitely 82.
