Video Transcript
What is the simplest form of the quadratic equation whose roots are 13 over two and five-thirds?
Well, when we talk about roots, it’s another word for solutions. So what we’re looking for is a quadratic equation whose solutions are 13 over two and five-thirds. Well, let’s say that our solutions are for 𝑥. So we know that 𝑥 is equal to 13 over two or 𝑥 is equal to five-thirds. But what we want to do is make each of these equal to zero cause this would help us to find the factors of our quadratic equation.
So therefore, we’re gonna subtract 13 over two from each side. And what we get is 𝑥 minus 13 over two is equal to zero. And then if we do the same to the other side, what we’ve got here is 𝑥 equals five over three. Well, if we subtract five over three from each side, we’re gonna get 𝑥 minus five over three or five-thirds is equal to zero.
So now what we want to do to make things more simple is to remove the fractions. And so we’re gonna do that on the left-hand side by multiplying through by two. And if we do that and we multiply each term by two, we’re gonna get two 𝑥 minus 13 is equal to zero. And if we do that on the right-hand side, but this time multiplying through by three to remove the fraction, we multiply each term by three. We’re gonna get three 𝑥 minus five is equal to zero.
So therefore, we know if the possible solutions are 13 over two and five over three, then we can rewrite our quadratic in a fully factored form as two 𝑥 minus 13 multiplied by three 𝑥 minus five is equal to zero. And that’s because if either of these parentheses are equal to zero, then the end result will be zero. So it would be correct.
So now all that’s left to do is distribute across our parentheses. So we’re gonna multiply first of all two 𝑥 by three 𝑥, which is gonna give us six 𝑥 squared. Then two 𝑥 by negative five, which gives us negative 10𝑥. Then negative 13 multiplied by three 𝑥, which gives us negative 39𝑥. So negative 13 multiplied by negative five gives us positive 65. So that’s great. So now we’ve got 16𝑥 squared minus 10𝑥 minus 39𝑥 plus 65 equals zero.
So the last stage is to simplify. And when we do, we get six 𝑥 squared minus 49𝑥 plus 65 equals zero. So therefore, we can say that this is the simplest form of the quadratic equation whose roots are 13 over two and five-thirds.
