Video: Forming Quadratic Equations in the Simplest Form Using the Relation between Them and Their Roots

If ๐ฟ and ๐‘€ are roots of the equation ๐‘ฅยฒ โˆ’ 19๐‘ฅ + 9 = 0, find, in its simplest form, the quadratic equation whose roots are ๐ฟ โˆ’ 2 and ๐‘€ โˆ’ 2.

09:39

Video Transcript

If ๐ฟ and ๐‘€ are roots of the equation ๐‘ฅ squared minus 19๐‘ฅ plus nine equals zero, find in its simplest form the quadratic equation whose roots are ๐ฟ minus two and ๐‘€ minus two.

Before we start this one, everyone needs to take a deep breath. It is definitely meticulous, but letโ€™s dive in anyway! ๐‘ฅ squared minus 19๐‘ฅ plus nine equals zero is our equation, and we need the roots of this equation. When I look at the values nine and negative 19, I realize that it doesnโ€™t have a simple factored solution. Weโ€™ll need to use a formula to find the roots here: negative ๐‘ plus or minus the square root of ๐‘ squared minus four ๐‘Ž๐‘ all over two ๐‘Ž.

๐‘Ž is the coefficient of ๐‘ฅ squared; for us, thatโ€™s one. ๐‘ is the coefficient of ๐‘ฅ, negative 19 for us. And ๐‘ is the constant value, nine. Weโ€™ll plug this in. Negative ๐‘ would be negative of negative 19, positive 19, plus or minus the square root of negative 19 squared, the four ๐‘Ž๐‘ becomes four times one times nine, all over two times one. Negative 19 squared is 361 minus 36. And then weโ€™ll divide each of these terms separately by two.

To simplify even further, 19 over two plus or minus the square root of 325 over two. We can simplify the square root of 325, but we donโ€™t have to. For now, letโ€™s leave it like this, the square root of 325 over two. Remember that weโ€™re going to call the roots of this equation ๐ฟ and ๐‘€. Weโ€™ll set ๐ฟ equal to 19 over two plus the square root of 325 over two. And that would make ๐‘€ 19 over two minus the square root of 325 over two.

And now weโ€™re basically finished with this first equation. But we want to use this information, use the roots ๐ฟ and ๐‘€, to help us write a new equation: ๐ฟ minus two would be equal to 19 over two plus the square root of 325 over two minus two. To solve this, letโ€™s write the whole number two as a fraction. The whole number two is equal to four over two written in fraction form. Weโ€™ll just move the equation around to group like terms. 19 minus four equals 15, and that makes ๐ฟ minus two equal to 15 over two plus the square root of 325 over two.

We need to do the same thing with ๐‘€. 19 over two minus the square root of 325 over two minus four over two. Regroup to say 19 over two minus four over two minus the square root of 325 over two. And that makes ๐‘€ minus two equal to 15 over two minus the square root of 325 over two. Our new roots, our new ๐‘ฅ values, would look like this. But remember, when weโ€™re working to find the equation, weโ€™ll need to set these roots equal to zero.

Following here with our ๐ฟ minus two root. Iโ€™m going to subtract 15 over two and the square root of 325 over two from both sides of the equation. On the right side, everything cancels out, leaving us with zero. And on the left side, we have ๐‘ฅ minus 15 over two minus the square root of 325 over two. Weโ€™ll just hold this term here for a second while we work on the left term. Iโ€™m going to subtract 15 over two from both sides of the equation. After I do that, Iโ€™ll have the equation ๐‘ฅ minus 15 over two equals negative the square root of 325 over two.

Remember that weโ€™re trying to set this equation equal to zero. If I add the square root of 325 over two to the left side and to the right side, the right side is equal to zero and the left side is equal to ๐‘ฅ minus 15 over two plus the square root of 325 over two. Now if we just took these two sets of parentheses and wrote them next to each other, that would be the quadratic equation whose roots are ๐ฟ two and ๐‘€ two. But it wouldnโ€™t be in its simplest form. And that means we still have some work to do.

Iโ€™m going to have to erase some of this to make more room. So now would be a good time to pause and copy it down if you want to see some of this later. What weโ€™re gonna do now is take the two sets of parentheses and multiply them with each other. We want to multiply ๐‘ฅ minus 15 over two minus the square root of 325 over two times ๐‘ฅ minus 15 over two plus the square root of 325 over two. Thatโ€™s a mouthful! But weโ€™ll follow this same process we always do: weโ€™ll FOIL.

First, ๐‘ฅ times ๐‘ฅ equals ๐‘ฅ squared. ๐‘ฅ times negative 15 over two equals negative 15 over two ๐‘ฅ. ๐‘ฅ times the square root of 325 over two equals the square root of 325 over two ๐‘ฅ. Now weโ€™ll move to our negative 15 over two term, negative 15 over two ๐‘ฅ, and then negative 15 over two times negative 15 over two, I just wanna write that as negative 15 over two squared for now. Negative 15 over two times the square root of 325 over two. Before I keep going, Iโ€™m gonna clear some more of this information off to the right.

Okay, for this term Iโ€™m actually not going to multiply all of this out. And youโ€™ll see why in a minute. Iโ€™m just gonna leave it as negative 15 times the square root of 325 over two times two. Weโ€™re now gonna move on to this last term, negative square root of 325 over two ๐‘ฅ. This time it will say negative 15 over two times negative square root of 325 over two. And finally, negative square root of 325 over two times positive square root of 325 over two. Now we have some simplifying to do.

Our ๐‘ฅ squared is good to go. Negative 15 over two ๐‘ฅ can be added to negative 15 over two ๐‘ฅ. That would be negative 30 over two ๐‘ฅ. Weโ€™ll just check those off. The next thing I see is the square root of 325 over two ๐‘ฅ, thatโ€™s a positive term, and then the negative term, negative square root of 325 over two ๐‘ฅ. These cancel each other out. The next thing, negative 15 over two squared. Negative 15 squared is 225. Two squared equals four. Weโ€™ve taken care of that term. The next thing, negative 15 over two times the square root of 325 over two and negative 15 over two times negative square root of 325 over two.

These are opposites of each other. The one on the right is positive; the one on the left is negative. They cancel each other out. Our final term, negative square root of 325 over two times positive square root of 325 over two. Positive times a negative is a negative. The square root of 325 times the square root of 325 equals 325, and two times two equals four. Can we reduce this any further? Well, we can simplify negative 30 over two. Negative 30 over two is equal to negative 15๐‘ฅ. 225 over four minus 325 over four equals negative 100 over four. 100 divided by four equals 25.

Our new equation that weโ€™ve derived is ๐‘ฅ squared minus 15๐‘ฅ minus 25. The equation whose roots are ๐ฟ minus two and ๐‘€ minus two in its simplest form, ๐‘ฅ squared minus 15๐‘ฅ minus 25. But if youโ€™re wondering what on earth we did, let me encourage you. Watch it again, and then try it for yourself. Actually write down each of these steps. We pretty much only did basic algebra, but we dealt with very difficult terms because of those square roots in there. The steps are still the same though, even when you work with irrational numbers. You can do it too!

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