Video Transcript
Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 squared minus two 𝑥 plus 20 equals zero, find, in its simplest form, the quadratic equation whose roots are two and 𝐿 squared plus 𝑀 squared.
We recall that any quadratic equation of the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero — which has two roots, 𝑟 sub one and 𝑟 sub two — then the sum of the roots is equal to negative 𝑏 over 𝑎 and the product of the two roots is equal to 𝑐 over 𝑎. We are given the quadratic equation 𝑥 squared minus two 𝑥 plus 20 equals zero. Therefore, 𝑎 is equal to one, 𝑏 is equal to negative two, and 𝑐 is equal to 20. The roots of this equation are 𝐿 and 𝑀. Therefore, 𝐿 plus 𝑀 is equal to negative negative two over one. This is equal to two. The product of the two roots, 𝐿 multiplied by 𝑀, is equal to 20 over one. This is equal to 20.
Using this information, we need to find another quadratic equation whose roots are two and 𝐿 squared plus 𝑀 squared. In this equation, two plus 𝐿 squared plus 𝑀 squared must equal negative 𝑏 over 𝑎 and two multiplied by 𝐿 squared plus 𝑀 squared equals 𝑐 over 𝑎, where 𝑎, 𝑏, and 𝑐 are unknowns we need to calculate. We recall that expanding 𝐿 plus 𝑀 all squared gives us 𝐿 squared plus two 𝐿𝑀 plus 𝑀 squared. Subtracting two 𝐿𝑀 from both sides of this equation tells us that 𝐿 squared plus 𝑀 squared is equal to 𝐿 plus 𝑀 all squared minus two 𝐿𝑀. We notice that the expression on the right-hand side is contained in both of our equations. We also notice that 𝐿𝑀 is equal to 20 and 𝐿 plus 𝑀 is equal to two.
Substituting in these values, 𝐿 squared plus 𝑀 squared is equal to two squared minus two multiplied by 20. The left-hand side simplifies to negative 36. We can now substitute this value into both of our equations. Negative 𝑏 over 𝑎 is equal to two plus negative 36. This is equal to negative 34. Two multiplied by negative 36 is equal to 𝑐 over 𝑎. 𝑐 over 𝑎 is therefore equal to negative 72. As both negative 34 and negative 72 are integers, we can let 𝑎 equal one. This means that negative 𝑏 is equal to negative 34, so 𝑏 equals 34. 𝑐 is equal to negative 72. The quadratic equation whose roots are two and 𝐿 squared plus 𝑀 squared is 𝑥 squared plus 34𝑥 minus 72 is equal to zero.
