# Video: Forming Quadratic Equations in the Simplest Form given Their Roots

Given that πΏ and π are the roots of the equation π₯Β² β 2π₯ + 20 = 0, find, in its simplest form, the quadratic equation whose roots are 2 and πΏΒ² + πΒ².

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### Video Transcript

Given that πΏ 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 πΏ squared plus π squared.

We recall that any quadratic equation of the form ππ₯ squared plus ππ₯ 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 πΏ and π. Therefore, πΏ plus π is equal to negative negative two over one. This is equal to two. The product of the two roots, πΏ 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 πΏ squared plus π squared. In this equation, two plus πΏ squared plus π squared must equal negative π over π and two multiplied by πΏ squared plus π squared equals π over π, where π, π, and π are unknowns we need to calculate. We recall that expanding πΏ plus π all squared gives us πΏ squared plus two πΏπ plus π squared. Subtracting two πΏπ from both sides of this equation tells us that πΏ squared plus π squared is equal to πΏ plus π all squared minus two πΏπ. We notice that the expression on the right-hand side is contained in both of our equations. We also notice that πΏπ is equal to 20 and πΏ plus π is equal to two.

Substituting in these values, πΏ 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 πΏ squared plus π squared is π₯ squared plus 34π₯ minus 72 is equal to zero.