### Video Transcript

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

We recall that for any quadratic equation in the form ππ₯ squared plus ππ₯ plus π equals zero, then the sum of our roots is equal to negative π over π and the product of the roots is equal to π over π. Weβre initially given the equation π₯ squared minus six π₯ minus four is equal to zero such that π is equal to one, π is equal to negative six, and π is equal to negative four.

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

Distributing the parentheses on the left-hand side gives us πΏπ plus six πΏ plus six π plus 36. This is equal to negative four. We can subtract 36 from both sides such that πΏπ plus six πΏ plus six π is equal to negative 40. We can factor out six from six πΏ plus six π. This means that πΏπ plus six multiplied by πΏ plus π is equal to negative 40. We already know that πΏ plus π is equal to negative six. This means that πΏπ 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 πΏπ is equal to negative four.

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

We already know the values of πΏπ and πΏ plus π. Negative π over π is, therefore, equal to negative four multiplied by negative six. This is equal to 24. Using the product of roots, πΏ squared π multiplied by π squared πΏ is equal to π over π. Using our laws of exponents or indices, the left-hand side simplifies to πΏ cubed π cubed. This in turn can be rewritten as πΏπ all cubed. As πΏπ is equal to negative four, π over π 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 π over π was equal to 24 and π over π was equal to negative 64. As both of these are integers, we can let π equal one. This means that negative π is equal to 24, and π is equal to negative 24. π is equal to negative 64.

The quadratic equation whose roots are πΏ squared π and π squared πΏ is π₯ squared minus 24π₯ minus 64 equals zero.