Question Video: Forming Quadratic Equation in the Simplest Form given Its Roots Mathematics • 10th Grade

Find, in its simplest form, the quadratic equation whose roots are π + 3π and π β 3π.

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Video Transcript

Find, in its simplest form, the quadratic equation whose roots are π plus three π and π minus three π.

Weβre given two roots. Sometimes we call the roots the solutions. We have a solution at π₯ equals π plus three π and a solution at π₯ equals π minus three π. To take the roots and find an equation, we want to set both of these equations equal to zero. We can subtract π from both sides. And then, we have π₯ minus π equals three π. So we subtract three π from both sides. And weβll have π₯ minus π minus three π equal zero. Weβll do the same thing on the other side: subtract π. But this time, we need to add three π to both sides. So on the left, we subtract π and we add three π. And we get π₯ minus π plus three π 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 minus three π times π₯ minus π plus three π. And so, now, we just need to expand. π₯ times π₯ equals π₯ squared. π₯ times negative π equals negative ππ₯. π₯ times three π equals positive three ππ₯. When we move on to our middle term, we have negative π times π₯, which is negative ππ₯. Negative π times negative π is positive π squared. Negative π times positive three π is negative three ππ. We have our final term negative three π times π₯ is negative three ππ₯. Negative three π times negative π is positive three ππ. And negative three π times positive three π is negative nine π squared all equal to zero.

Now, we need to carefully group the like terms. Thereβs only one π₯ squared term. But we have two negative ππ₯ terms. We can combine them by calling that negative two ππ₯. We have a positive three ππ₯ and a negative three ππ₯. When theyβre combined, they equal zero. We only have one π squared term. So we just bring it down. We have a negative three ππ and a positive three ππ. When theyβre combined, they equal zero. And thereβs only one π squared term. Thatβs negative nine π 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π₯ plus π squared minus nine π squared equal zero.