Given that 𝐿 and 𝑀 are the roots of the equation two 𝑥 squared minus 10𝑥 plus one equals zero, find, in its simplest form, the quadratic equation whose roots are 𝐿 over three and 𝑀 over three.
We begin by recalling some key facts about a quadratic equation written in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, where 𝑎, 𝑏, and 𝑐 are constants and 𝑎 is nonzero. If the two roots of the quadratic are 𝑟 sub one and 𝑟 sub two, their sum is equal to negative 𝑏 over 𝑎. The product of the two roots, 𝑟 one multiplied by 𝑟 two, is equal to 𝑐 over 𝑎.
In this question, we are given the equation two 𝑥 squared minus 10𝑥 plus one equals zero. The values of 𝑎, 𝑏, and 𝑐 here are two, negative 10, and one. We are also told that 𝐿 and 𝑀 are the two roots of the equation. Therefore, the sum of these, 𝐿 plus 𝑀, is equal to negative negative 10 over two. This is equal to five. The product of the roots 𝐿 multiplied by 𝑀 is equal to one over two or one-half. We can now use these values to help find the quadratic equation whose roots are 𝐿 over three and 𝑀 over three.
Let’s begin by considering the sum of these roots. We have 𝐿 over three plus 𝑀 over three. As the denominators are the same, we can simply add the numerators, giving us 𝐿 plus 𝑀 over three. We have already worked out that 𝐿 plus 𝑀 is equal to five. This means that the sum of the roots is equal to five over three or five-thirds. Negative 𝑏 over 𝑎 is equal to five-thirds.
We will now consider the product of our two roots. We need to multiply 𝐿 over three and 𝑀 over three. We do this by multiplying the numerators and denominators separately, giving us 𝐿𝑀 over nine. As 𝐿𝑀 is equal to one-half, we have a half over nine or a half divided by nine. This is equal to one eighteenth. The product of our roots 𝑐 over 𝑎 is equal to one eighteenth.
We can now solve these two equations to find the values of 𝑎, 𝑏, and 𝑐. We note that the denominators on the right-hand side are different. However, on the left-hand side of both our equations, the denominator is 𝑎. We can therefore multiply the numerator and denominator of the right-hand side of our first equation by six. Five over three or five-thirds is the same as 30 over 18.
Our next step is to let 𝑎 equal 18. From the first equation, this means that negative 𝑏 is equal to 30, which means that 𝑏 is equal to negative 30. From the second equation, if 𝑐 over 𝑎 is equal to one over 18 and 𝑎 equals 18, then 𝑐 must be equal to one. The quadratic equation whose roots are 𝐿 over three and 𝑀 over three is 18𝑥 squared minus 30𝑥 plus one equals zero.