Question Video: Forming a Quadratic Equation in the Simplest Form given Its Roots Mathematics

Video Transcript

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

We recall that for any quadratic equation in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then the sum of our roots is equal to negative 𝑏 over 𝑎 and the product of the roots is equal to 𝑐 over 𝑎. We’re initially given the equation 𝑥 squared minus six 𝑥 minus four is equal to zero such that 𝑎 is equal to one, 𝑏 is equal to negative six, and 𝑐 is equal to negative four.

𝐿 plus six and 𝑀 plus six are the roots of this equation. This means that 𝐿 plus six plus 𝑀 plus six is equal to negative negative six over one. The left-hand side of our equation simplifies to 𝐿 plus 𝑀 plus 12. This is equal to six. We can then subtract 12 from both sides of this equation such that 𝐿 plus 𝑀 is equal to negative six. Using the product of roots, we know that 𝐿 plus six multiplied by 𝑀 plus six is equal to negative four over one.

Distributing the parentheses on the left-hand side gives us 𝐿𝑀 plus six 𝐿 plus six 𝑀 plus 36. This is equal to negative four. We can subtract 36 from both sides such that 𝐿𝑀 plus six 𝐿 plus six 𝑀 is equal to negative 40. We can factor out six from six 𝐿 plus six 𝑀. This means that 𝐿𝑀 plus six multiplied by 𝐿 plus 𝑀 is equal to negative 40. We already know that 𝐿 plus 𝑀 is equal to negative six. This means that 𝐿𝑀 plus six multiplied by negative six is equal to negative 40. Six multiplied by negative six is negative 36. We can then add 36 to both sides of this equation, giving us 𝐿𝑀 is equal to negative four.

We now have two equations linking 𝐿 and 𝑀. 𝐿 and 𝑀 have a sum of negative six and a product of negative four. We need to find a new quadratic equation where the roots are 𝐿 squared 𝑀 and 𝑀 squared 𝐿. 𝐿 squared 𝑀 plus 𝑀 squared 𝐿 must be equal to negative 𝑏 over 𝑎, where 𝑎 and 𝑏 are values we need to calculate. We can factor out an 𝐿 and an 𝑀 on the left-hand side such that 𝐿 squared 𝑀 plus 𝑀 squared 𝐿 is equal to 𝐿𝑀 multiplied by 𝐿 plus 𝑀.

We already know the values of 𝐿𝑀 and 𝐿 plus 𝑀. Negative 𝑏 over 𝑎 is, therefore, equal to negative four multiplied by negative six. This is equal to 24. Using the product of roots, 𝐿 squared 𝑀 multiplied by 𝑀 squared 𝐿 is equal to 𝑐 over 𝑎. Using our laws of exponents or indices, the left-hand side simplifies to 𝐿 cubed 𝑀 cubed. This in turn can be rewritten as 𝐿𝑀 all cubed. As 𝐿𝑀 is equal to negative four, 𝑐 over 𝑎 is equal to negative four cubed. This is equal to negative 64.

We will now clear some space in order to complete this question. We found that negative 𝑏 over 𝑎 was equal to 24 and 𝑐 over 𝑎 was equal to negative 64. As both of these are integers, we can let 𝑎 equal one. This means that negative 𝑏 is equal to 24, and 𝑏 is equal to negative 24. 𝑐 is equal to negative 64.

The quadratic equation whose roots are 𝐿 squared 𝑀 and 𝑀 squared 𝐿 is 𝑥 squared minus 24𝑥 minus 64 equals zero.