Video Transcript
Given that 𝐿 and 𝑀 are the roots of the equation two 𝑥 squared minus nine 𝑥 plus nine equals zero, find, in its simplest form, the quadratic equation whose roots are 𝐿 over 𝑀 and 𝑀 over 𝐿.
We recall that for any quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then the sum of the roots is equal to negative 𝑏 over 𝑎 and the product of the roots is equal to 𝑐 over 𝑎. In our equation, two 𝑥 squared minus nine 𝑥 plus nine equals zero, 𝑎 is equal to two, 𝑏 is equal to negative nine, and 𝑐 is equal to nine.
We are told that the roots are 𝐿 and 𝑀. Therefore, 𝐿 plus 𝑀 is equal to negative negative nine over two. This simplifies to nine over two. The product of the roots 𝐿 multiplied by 𝑀 is also equal to nine over two.
We are asked to find the quadratic whose roots are 𝐿 over 𝑀 and 𝑀 over 𝐿. Let’s begin by considering the product of these roots. 𝐿 over 𝑀 multiplied by 𝑀 over 𝐿 is equal to 𝑐 over 𝑎, where 𝑐 and 𝑎 are values we need to calculate. The left-hand side simplifies to 𝐿𝑀 over 𝐿𝑀, which is equal to one. 𝑐 over 𝑎 is equal to one.
Let’s now consider the sum of the roots. 𝐿 over 𝑀 plus 𝑀 over 𝐿 is equal to negative 𝑏 over 𝑎. The left-hand side can be simplified to 𝐿 squared plus 𝑀 squared divided by 𝑀𝐿. We multiply the numerator and denominator of the first fraction by 𝐿 and the second fraction by 𝑀. The denominator can be rewritten as 𝐿𝑀, which we know is equal to nine over two.
We recall that 𝐿 plus 𝑀 all squared is equal to 𝐿 squared plus two 𝐿𝑀 plus 𝑀 squared. Subtracting two 𝐿𝑀 from both sides, 𝐿 squared plus 𝑀 squared can be rewritten as 𝐿 plus 𝑀 all squared minus two 𝐿𝑀. This means that negative 𝑏 over 𝑎 is equal to 𝐿 plus 𝑀 squared minus two 𝐿𝑀 all divided by 𝐿𝑀.
We could now substitute nine over two into this equation. Negative 𝑏 over 𝑎 is equal to nine over two squared minus two multiplied by nine over two all divided by nine over two. This is equal to five over two. Negative 𝑏 over 𝑎 is equal to five over two or five-halves.
As 𝑎 is on the denominator of both of the left-hand sides, we can rewrite one as two over two. We can now let 𝑎 equal two, which also means that 𝑐 is equal to two. Negative 𝑏 is equal to five, which means that 𝑏 is equal to negative five. The quadratic equation whose roots are 𝐿 over 𝑀 and 𝑀 over 𝐿 is two 𝑥 squared minus five 𝑥 plus two equals zero.
