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# Solving quadratics by taking square roots: challenge

Sal solves challenging quadratic equations like (4x+1)²-8=0 by taking the square root of both sides. Created by Sal Khan and CK-12 Foundation.

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• How would you solve -8n(10n-1)=0 using the zero-product property? I'm not really getting exactly how to use this process. Could someone help me with this? • The point of the zero-product property is this:
If two or more factors are multiplied together to make 0, then one of the factors must = 0.
Think about it, if you want to make 0 by multiplying, you have to have a 0 as a factor:
0 * 8 = 0
125 * 0 = 0
1/4 * 0 = 0
2 * 3 * 4 * 5 * 6 * 0 = 0

So, if you know that a product is 0, then one of the factors MUST be 0.
In this problem:
-8n(10n-1)=0
The product is 0.
The two factors are: -8n and 10n-1
At least one of the factors must be =0.
So, set each one equal to 0.
-8n = 0
-8n/-8 = 0/-8
`n = 0`

or

10n-1 = 0
10n -1 +1 = 0 + 1
10n = 1
`n = 1/10`

`n = 0 OR n = 1/10`
• How come quadratics have 2 answers that both work only? Why not only one or 3? Is that the structure of quadratic equations to get 2 answers only? • Yes, you are right. The quadratic equation is structured so that you end up with two roots, or solutions. This is because in the quadratic formula (-b+-√b^2-4ac) / 2a, it includes a radical. When taking the square root of something, you can have a positive square root (the principle square root) or the negative square root. For example: √25 = 5, but the √25 = -5 as well because -5*-5 is 25 also. That's where we get the "plus or minus" in the quadratic formula. So in one solution, you add the radical (which is the solution with the principle square root) and in another, you subtract (hence the negative square root). Hope this helped! :)
• For (4x+1) squared why don't we do the negative square root and do only the positive when we do both for the other side? • You do not have to put the plus or minus on the variable side because it is redundant. You already have the positives
4x+1 = 2sqrt(2)..... and 4x+1 = -2sqrt(2), right?

Now think about what would happen if you looked at the negatives, you have
-(4x+1) = 2sqrt(2) which, if you multiply both sides by -1 gives you 4x+1 = -2sqrt(2)
and the other
-(4x+1) = -2sqrt(2) again, multiply by -1 and you have
4x+1 = 2sqrt(2).

So, putting the plus and the minus on the left side also, gives you expressions equivalent to the expressions if you put it only on the right side. No point in making yourself extra work for no good reason, right?
• At , what is the step-by-step procedure to solving (x-8)(x-2)=0 when solving for x? How did Sal immediately know that the answer would be x=8 or x=2? The only way I see him working it out is if he divided the expression up into two parts like so; (x-8)=0 and (x-2)=0 and then solving for x. If so, why is this allowed? It doesn't make intuitive sense. • Basically the expression (x+8)(x+2)=0 means that one of those two factor [you have to look at (x+8) as one factor and at (x+2) as another factor] must be equal to zero to satisfy the equation.
Either (x+8) must be zero or (x+2) must be zero, or both. Therefore if x+8=0 then x = -8 and this is one valid solution, similarly if x+2=0 then x= -2 and this is the second valid solution.

When you tackle problems like this remember that, for a product to be equal to zero there must be a factor which is 0. Hope it helped
• General Question. Why are we so concerned with the so called "solutions" of these functions? There can be many other values of x in f(x)=ax^2+bx+c which give a valid output. Why do we need to find which value/s of x give zero as output? • Why all this trouble to find where functions are zero?
It tells us something about the behavior of the function, for example, once we know where the zeros are, we can look on either side of the zero values and find out if the function is positive or negative, which in turn gives clues to where maximum and minimum values may be lurking. Later, in calculus, you will learn to take the derivatives of functions and try to find where the derivatives are zero as well, and when we find them, it tells us even more about the behavior of the function such as how fast it changes from negative to positive or vice-versa, and where the maximum/minimum value is or if it even exists.

All this procedure is part of a mathematical process called analysis, which is really what math is all about. You see, we can model much of the world via functions and this type of analysis helps us understand the model, which in turns helps us understand the nature of what we are trying to model, which gives us a better understanding of nature itself.

Right now you are honing your skills at algebra, which will never leave your side; no matter how far up the math ladder you want to climb, algebra is there. With some of these "honing" skills, you may not be able to see the reasons, whys and wherefores at the moment, but that will come in time.

Keep studying!
• I still don't get why it is plus or minus the square root. • It would be because the Parabola of a quadratic equation crosses the x- axis in both the negative values and the positive values. In other words the x-intercepts are both positive and negative. Note that sometimes the square root is so big that the plus or minus does not have an effect on the side. That is why some parabolas are on the right side and do not have a negative solution. Hope this helped!
• Why do you have to put a + and - when you square root something? • I don't understand what the positive square root and negative square root are (also the +/- signs before the square root symbol) .....so like when someone says you can have the positive square root of this or the negative...what does that mean?? • does anyone have an idea how the formula -b+-root b*-4ac/2a is derived ? • how would u solve this: (x+2)^2=10 • There are two possible answers here, namely:
`Answer using the positive + square root:x=-2+√10`
`Answer using the negative - square root:x=-2-√10`
Solution:
(x+2)²=10 → this is our original equation
√(x+2)²=√10 → let's take the square roots of both sides
x+2=±√10 → this is what we'll get, then I'll subtract 2 from both sides

`taking the positive square root.....`
x=√10-2 → this is what I got, and now I'll rearrange the order
x=-2+√10 → I rearranged the order and there we have it!

`or taking the negative square`` root.....`
x=-√10-2 → this is what I got, and now I'll rearrange the order
x=-2-√10 → I rearranged the order and there we have it again!