### Video Transcript

Given that πΏ and π are the roots of the equation two π₯ squared minus 21π₯ plus four equals zero, find, in its simplest form, the quadratic equation whose roots are two πΏ and two π.

We begin by recalling the properties of a quadratic equation written in the form ππ₯ squared plus ππ₯ plus π equals zero, where π, π, and π are constants and π is nonzero. If this equation has roots π sub one and π sub two, then the sum of these is equal to negative π over π. The product of the two roots, π one multiplied by π two, is equal to π over π. In this question, weβre given the equation two π₯ squared minus 21π₯ plus four equals zero. The value of π is equal to two, π is equal to negative 21, and π is equal to four.

We are told that the two roots are πΏ and π. Therefore, πΏ plus π is equal to negative negative 21 over two. This simplifies to 21 over two. The product of the roots πΏ multiplied by π is equal to four over two. This is equal to two. We have been asked to find the quadratic equation whose roots are two πΏ and two π. The sum of these roots is equal to two πΏ plus two π. These terms have a common factor of two, so this can be rewritten as two multiplied by πΏ plus π. We know that πΏ plus π is equal to 21 over two. Therefore, the sum of the roots equals two multiplied by 21 over two. This is equal to 21. Negative π over π is therefore equal to 21.

The product of our two roots is equal to two πΏ multiplied by two π. This is equal to four πΏπ. As πΏπ is equal to two, we have four multiplied by two. We can therefore conclude that π over π is equal to eight. Using the two equations negative π over π equals 21 and π over π equals eight, we can calculate values for π, π, and π. This will in turn give us the quadratic equation whose roots are two πΏ and two π. As both 21 and eight are integers, we can let π equal one. This means that negative π is equal to 21 and π is therefore equal to negative 21. If π is equal to one from the second equation, π is equal to eight.

The quadratic equation whose roots are two πΏ and two π is π₯ squared minus 21π₯ plus eight is equal to zero.