Video: Forming Quadratic Equation in the Simplest Form given Its Roots

Find, in its simplest form, the quadratic equation whose roots are π‘š + 3𝑛 and π‘š βˆ’ 3𝑛.

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Video Transcript

Find, in its simplest form, the quadratic equation whose roots are π‘š plus three 𝑛 and π‘š minus three 𝑛.

We’re given two roots. Sometimes we call the roots the solutions. We have a solution at π‘₯ equals π‘š plus three 𝑛 and a solution at π‘₯ equals π‘š 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 π‘š from both sides. And then, we have π‘₯ minus π‘š equals three 𝑛. So we subtract three 𝑛 from both sides. And we’ll have π‘₯ minus π‘š minus three 𝑛 equal zero. We’ll do the same thing on the other side: subtract π‘š. But this time, we need to add three 𝑛 to both sides. So on the left, we subtract π‘š and we add three 𝑛. And we get π‘₯ minus π‘š 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 π‘₯ minus π‘š minus three 𝑛 times π‘₯ minus π‘š plus three 𝑛. And so, now, we just need to expand. π‘₯ times π‘₯ equals π‘₯ squared. π‘₯ times negative π‘š equals negative π‘šπ‘₯. π‘₯ times three 𝑛 equals positive three 𝑛π‘₯. When we move on to our middle term, we have negative π‘š times π‘₯, which is negative π‘šπ‘₯. Negative π‘š times negative π‘š is positive π‘š squared. Negative π‘š times positive three 𝑛 is negative three π‘›π‘š. We have our final term negative three 𝑛 times π‘₯ is negative three 𝑛π‘₯. Negative three 𝑛 times negative π‘š is positive three π‘›π‘š. 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 π‘₯ squared term. But we have two negative π‘šπ‘₯ terms. We can combine them by calling that negative two π‘šπ‘₯. We have a positive three 𝑛π‘₯ and a negative three 𝑛π‘₯. When they’re combined, they equal zero. We only have one π‘š squared term. So we just bring it down. We have a negative three π‘›π‘š and a positive three π‘›π‘š. 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 π‘₯ squared minus two π‘šπ‘₯ plus π‘š squared minus nine 𝑛 squared equal zero.

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