### Video Transcript

Given that πΏ plus three and π plus three are the roots of the equation π₯ squared plus eight π₯ plus 12 is equal to zero, find in its simplest form the quadratic equation whose roots are πΏ and π.

In this question, we are being asked to work backwards from knowing the roots of a quadratic equation to finding the quadratic equation itself. Letβs consider how to do this. If the quadratic equation has roots πΏ and π, then it can be factorized as π₯ minus πΏ multiplied by π₯ minus π.

Now, letβs multiply out the brackets to see what the expanded form of the quadratic would look like in terms of πΏ and π. We would have π₯ squared minus πΏπ₯ minus ππ₯ plus πΏπ is equal to zero. The π₯ terms can be factorized giving π₯ squared minus πΏ plus ππ₯ plus πΏπ is equal to zero. Now, letβs look at the coefficient in its expanded form more closely.

The coefficient of π₯ is negative πΏ plus π and πΏ plus π is the sum of the two roots. The constant term is plus πΏπ which is plus the product of the roots. What this means is that if we can find the sum of πΏ and π and the product of πΏ and π, then weβll be able to determine the quadratic equation whose roots they are.

Weβve been told in the question about a different quadratic equation π₯ squared plus eight π₯ plus 12 is equal to zero, whose roots are πΏ plus three and π plus three. Letβs see how this helps. We can first express this quadratic slightly differently by writing the middle term positive eight π₯ as negative negative eight π₯. Using what weβve just determined, this means that for this quadratic, the sum of its roots is negative eight and the product of its roots is 12.

The roots of this quadratic remember are πΏ plus three and π plus three. So we can form some equations. Firstly, as the sum of the roots is negative eight, we have the equation πΏ plus three plus π plus three is equal to negative eight. Secondly, as the product of the roots is 12, we have the equation πΏ plus three multiplied by π plus three is equal to 12.

We can now use these two equations to find the sum and product of πΏ and π so that we can substitute them back into the equation of the quadratic weβre looking to find. The first equation simplifies by adding the two threes together to give πΏ plus π plus six is equal to negative eight. Subtracting six from both sides gives πΏ plus π is equal to negative 14. And now, weβve found the sum of πΏ and π. This means that weβll be able to determine the coefficient of π₯ in our quadratic equation. The sum of πΏ and π is negative 14, which means a coefficient of π₯ will be minus negative 14 or 14.

Now, letβs consider the second equation. Expanding the double brackets gives πΏπ plus three πΏ plus three π plus nine is equal to 12. Subtracting nine from both sides of the equation gives πΏπ plus three πΏ plus three π is equal to three. Remember weβre still looking to determine the product of these roots β πΏπ. So this equation looks like itβs going to be useful. We can factorize plus three πΏ plus three π to give πΏπ plus three lots of πΏ plus π is equal to three. Remember weβve already determined what the sum of πΏ and π is. Itβs negative 14, which means we can substitute this value into the equation.

We now have πΏπ plus three multiplied by negative 14 is equal to three. Three multiplied by negative 14 is negative 42. And if we add 42 to both sides, we find that πΏπ is equal to 45. So now, we also know the product of πΏ and π. And we can use this to find the constant term in our quadratic equation. Substituting the values we found for πΏ plus π and πΏπ into the quadratic equation, we have π₯ squared minus negative 14π₯ plus 45 is equal to zero.

Simplifying the coefficient for the π₯ terms gives our quadratic equation in its simplest form. The quadratic equation whose roots are πΏ and π is π₯ squared plus 14π₯ plus 45 is equal to zero.