Find, in its simplest form, the quadratic equation whose roots are 𝑚 plus three 𝑛 and 𝑚 minus three 𝑛.
We’re given two roots. Sometimes we call the roots the solutions. We have a solution at 𝑥 equals 𝑚 plus three 𝑛 and a solution at 𝑥 equals 𝑚 minus three 𝑛. To take the roots and find an equation, we want to set both of these equations equal to zero. We can subtract 𝑚 from both sides. And then, we have 𝑥 minus 𝑚 equals three 𝑛. So we subtract three 𝑛 from both sides. And we’ll have 𝑥 minus 𝑚 minus three 𝑛 equal zero. We’ll do the same thing on the other side: subtract 𝑚. But this time, we need to add three 𝑛 to both sides. So on the left, we subtract 𝑚 and we add three 𝑛. And we get 𝑥 minus 𝑚 plus three 𝑛 equal zero.
We now have two factors of the quadratic equation. And we multiply these factors together to expand the equation. This quadratic is 𝑥 minus 𝑚 minus three 𝑛 times 𝑥 minus 𝑚 plus three 𝑛. And so, now, we just need to expand. 𝑥 times 𝑥 equals 𝑥 squared. 𝑥 times negative 𝑚 equals negative 𝑚𝑥. 𝑥 times three 𝑛 equals positive three 𝑛𝑥. When we move on to our middle term, we have negative 𝑚 times 𝑥, which is negative 𝑚𝑥. Negative 𝑚 times negative 𝑚 is positive 𝑚 squared. Negative 𝑚 times positive three 𝑛 is negative three 𝑛𝑚. We have our final term negative three 𝑛 times 𝑥 is negative three 𝑛𝑥. Negative three 𝑛 times negative 𝑚 is positive three 𝑛𝑚. And negative three 𝑛 times positive three 𝑛 is negative nine 𝑛 squared all equal to zero.
Now, we need to carefully group the like terms. There’s only one 𝑥 squared term. But we have two negative 𝑚𝑥 terms. We can combine them by calling that negative two 𝑚𝑥. We have a positive three 𝑛𝑥 and a negative three 𝑛𝑥. When they’re combined, they equal zero. We only have one 𝑚 squared term. So we just bring it down. We have a negative three 𝑛𝑚 and a positive three 𝑛𝑚. When they’re combined, they equal zero. And there’s only one 𝑛 squared term. That’s negative nine 𝑛 squared. So we bring that down all equal to zero.
And that means our quadratic equation in its simplest form is 𝑥 squared minus two 𝑚𝑥 plus 𝑚 squared minus nine 𝑛 squared equal zero.