Given that 𝐿 plus three and 𝑀 plus three are the roots of the equation 𝑥 squared plus eight 𝑥 plus 12 is equal to zero, find in its simplest form the quadratic equation whose roots are 𝐿 and 𝑀.
In this question, we are being asked to work backwards from knowing the roots of a quadratic equation to finding the quadratic equation itself. Let’s consider how to do this. If the quadratic equation has roots 𝐿 and 𝑀, then it can be factorized as 𝑥 minus 𝐿 multiplied by 𝑥 minus 𝑀.
Now, let’s multiply out the brackets to see what the expanded form of the quadratic would look like in terms of 𝐿 and 𝑀. We would have 𝑥 squared minus 𝐿𝑥 minus 𝑀𝑥 plus 𝐿𝑀 is equal to zero. The 𝑥 terms can be factorized giving 𝑥 squared minus 𝐿 plus 𝑀𝑥 plus 𝐿𝑀 is equal to zero. Now, let’s look at the coefficient in its expanded form more closely.
The coefficient of 𝑥 is negative 𝐿 plus 𝑀 and 𝐿 plus 𝑀 is the sum of the two roots. The constant term is plus 𝐿𝑀 which is plus the product of the roots. What this means is that if we can find the sum of 𝐿 and 𝑀 and the product of 𝐿 and 𝑀, then we’ll be able to determine the quadratic equation whose roots they are.
We’ve been told in the question about a different quadratic equation 𝑥 squared plus eight 𝑥 plus 12 is equal to zero, whose roots are 𝐿 plus three and 𝑀 plus three. Let’s see how this helps. We can first express this quadratic slightly differently by writing the middle term positive eight 𝑥 as negative negative eight 𝑥. Using what we’ve 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 𝐿 plus three and 𝑀 plus three. So we can form some equations. Firstly, as the sum of the roots is negative eight, we have the equation 𝐿 plus three plus 𝑀 plus three is equal to negative eight. Secondly, as the product of the roots is 12, we have the equation 𝐿 plus three multiplied by 𝑀 plus three is equal to 12.
We can now use these two equations to find the sum and product of 𝐿 and 𝑀 so that we can substitute them back into the equation of the quadratic we’re looking to find. The first equation simplifies by adding the two threes together to give 𝐿 plus 𝑀 plus six is equal to negative eight. Subtracting six from both sides gives 𝐿 plus 𝑀 is equal to negative 14. And now, we’ve found the sum of 𝐿 and 𝑀. This means that we’ll be able to determine the coefficient of 𝑥 in our quadratic equation. The sum of 𝐿 and 𝑀 is negative 14, which means a coefficient of 𝑥 will be minus negative 14 or 14.
Now, let’s consider the second equation. Expanding the double brackets gives 𝐿𝑀 plus three 𝐿 plus three 𝑀 plus nine is equal to 12. Subtracting nine from both sides of the equation gives 𝐿𝑀 plus three 𝐿 plus three 𝑀 is equal to three. Remember we’re still looking to determine the product of these roots — 𝐿𝑀. So this equation looks like it’s going to be useful. We can factorize plus three 𝐿 plus three 𝑀 to give 𝐿𝑀 plus three lots of 𝐿 plus 𝑀 is equal to three. Remember we’ve already determined what the sum of 𝐿 and 𝑀 is. It’s negative 14, which means we can substitute this value into the equation.
We now have 𝐿𝑀 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 𝐿𝑀 is equal to 45. So now, we also know the product of 𝐿 and 𝑀. And we can use this to find the constant term in our quadratic equation. Substituting the values we found for 𝐿 plus 𝑀 and 𝐿𝑀 into the quadratic equation, we have 𝑥 squared minus negative 14𝑥 plus 45 is equal to zero.
Simplifying the coefficient for the 𝑥 terms gives our quadratic equation in its simplest form. The quadratic equation whose roots are 𝐿 and 𝑀 is 𝑥 squared plus 14𝑥 plus 45 is equal to zero.