Video: Forming a Quadratic Equation in the Simplest Form Using the Relation between the Coefficient of a Quadratic Function and Its Roots

Video Transcript

Given that 𝐿 and 𝑀 are the roots of the equation 𝑥 squared minus 16𝑥 minus six equals zero, find, in its simplest form, the quadratic equation whose roots are 𝐿 plus 𝑀 and 𝐿𝑀.

We begin by recalling a couple of facts about any quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is nonzero. If the two roots of the equation are 𝑟 sub one and 𝑟 sub two, their sum 𝑟 sub one plus 𝑟 sub two is equal to negative 𝑏 over 𝑎. The product of the two roots 𝑟 sub one multiplied by 𝑟 sub two is equal to 𝑐 over 𝑎.

In this question, we are given the quadratic equation 𝑥 squared minus 16𝑥 minus six is equal to zero. This means that 𝑎 is equal to one, 𝑏 is equal to negative 16, and 𝑐 is equal to negative six. We are told that the two roots are 𝐿 and 𝑀. This means that 𝐿 plus 𝑀 is equal to negative negative 16 over one. This is equal to 16. The product of the roots 𝐿 multiplied by 𝑀 is equal to negative six over one. This is equal to negative six.

We need to find the quadratic equation whose roots are 𝐿 plus 𝑀 and 𝐿𝑀. The sum of these roots is equal to 𝐿 plus 𝑀 plus 𝐿𝑀. We already have values for both of these. They are equal to 16 and negative six. 16 plus negative six is the same as 16 minus six, which equals 10. This is equal to negative 𝑏 over 𝑎. The product of the roots is equal to 𝐿 plus 𝑀 multiplied by 𝐿𝑀. This is equal to 16 multiplied by negative six. Negative 96 is therefore equal to 𝑐 over 𝑎.

As both 10 and negative 96 are integers, we can let 𝑎 equal one. This means that 10 is equal to negative 𝑏. And 𝑏 is therefore equal to negative 10. If 𝑎 is equal to one, 𝑐 is equal to negative 96. The quadratic equation whose roots are 𝐿 plus 𝑀 and 𝐿𝑀 is 𝑥 squared minus 10𝑥 minus 96 is equal to zero.