Given that 𝐿 and 𝑀 are the roots of the equation two 𝑥 squared minus 21𝑥 plus four equals zero, find, in its simplest form, the quadratic equation whose roots are two 𝐿 and two 𝑀.
We begin by recalling the properties of 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 this equation has roots 𝑟 sub one and 𝑟 sub two, then the sum of these 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’re given the equation two 𝑥 squared minus 21𝑥 plus four equals zero. The value of 𝑎 is equal to two, 𝑏 is equal to negative 21, and 𝑐 is equal to four.
We are told that the two roots are 𝐿 and 𝑀. Therefore, 𝐿 plus 𝑀 is equal to negative negative 21 over two. This simplifies to 21 over two. The product of the roots 𝐿 multiplied by 𝑀 is equal to four over two. This is equal to two. We have been asked to find the quadratic equation whose roots are two 𝐿 and two 𝑀. The sum of these roots is equal to two 𝐿 plus two 𝑀. These terms have a common factor of two, so this can be rewritten as two multiplied by 𝐿 plus 𝑀. We know that 𝐿 plus 𝑀 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 𝑏 over 𝑎 is therefore equal to 21.
The product of our two roots is equal to two 𝐿 multiplied by two 𝑀. This is equal to four 𝐿𝑀. As 𝐿𝑀 is equal to two, we have four multiplied by two. We can therefore conclude that 𝑐 over 𝑎 is equal to eight. Using the two equations negative 𝑏 over 𝑎 equals 21 and 𝑐 over 𝑎 equals eight, we can calculate values for 𝑎, 𝑏, and 𝑐. This will in turn give us the quadratic equation whose roots are two 𝐿 and two 𝑀. As both 21 and eight are integers, we can let 𝑎 equal one. This means that negative 𝑏 is equal to 21 and 𝑏 is therefore equal to negative 21. If 𝑎 is equal to one from the second equation, 𝑐 is equal to eight.
The quadratic equation whose roots are two 𝐿 and two 𝑀 is 𝑥 squared minus 21𝑥 plus eight is equal to zero.