### Video Transcript

Given that πΏ and π are the roots of the equation π₯ squared minus two π₯ plus 20 equals zero, find, in its simplest form, the quadratic equation whose roots are two and πΏ squared plus π squared.

We recall that any quadratic equation of the form ππ₯ squared plus ππ₯ plus π equals zero β which has two roots, π sub one and π sub two β then the sum of the roots is equal to negative π over π and the product of the two roots is equal to π over π. We are given the quadratic equation π₯ squared minus two π₯ plus 20 equals zero. Therefore, π is equal to one, π is equal to negative two, and π is equal to 20. The roots of this equation are πΏ and π. Therefore, πΏ plus π is equal to negative negative two over one. This is equal to two. The product of the two roots, πΏ multiplied by π, 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 πΏ squared plus π squared. In this equation, two plus πΏ squared plus π squared must equal negative π over π and two multiplied by πΏ squared plus π squared equals π over π, where π, π, and π are unknowns we need to calculate. We recall that expanding πΏ plus π all squared gives us πΏ squared plus two πΏπ plus π squared. Subtracting two πΏπ from both sides of this equation tells us that πΏ squared plus π squared is equal to πΏ plus π all squared minus two πΏπ. We notice that the expression on the right-hand side is contained in both of our equations. We also notice that πΏπ is equal to 20 and πΏ plus π is equal to two.

Substituting in these values, πΏ squared plus π 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 π over π is equal to two plus negative 36. This is equal to negative 34. Two multiplied by negative 36 is equal to π over π. π over π is therefore equal to negative 72. As both negative 34 and negative 72 are integers, we can let π equal one. This means that negative π is equal to negative 34, so π equals 34. π is equal to negative 72. The quadratic equation whose roots are two and πΏ squared plus π squared is π₯ squared plus 34π₯ minus 72 is equal to zero.