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Lesson 3: Solving quadratics by taking square roots- Solving quadratics by taking square roots
- Solving quadratics by taking square roots
- Quadratics by taking square roots (intro)
- Solving quadratics by taking square roots examples
- Quadratics by taking square roots
- Solving quadratics by taking square roots: with steps
- Quadratics by taking square roots: strategy
- Quadratics by taking square roots: strategy
- Quadratics by taking square roots: with steps
- Solving simple quadratics review
- Solving quadratics by taking square roots: challenge
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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?(8 votes)
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/-8n = 0
or
10n-1 = 0
10n -1 +1 = 0 + 1
10n = 1n = 1/10
So, the answer is:n = 0 OR n = 1/10(22 votes)
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?(6 votes)
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! :)(10 votes)
- 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?(4 votes)
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?(6 votes)
At8:45, 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.(3 votes)
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(6 votes)
- 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?(3 votes)
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!(4 votes)
I still don't get why it is plus or minus the square root.(3 votes)
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!(3 votes)
Why do you have to put a + and - when you square root something?(3 votes)- Let's say we take the square root of 4. Both +2 x +2 and -2 x -2 = +4. Unless it is clear that a negative solution has no meaning, you need to show both solutions. (For example, if a floor is square and has an area of 64 square feet, a side could be either +8 feet of -8 feet. However, in this problem -8 feet doesn't have a meaning.)(3 votes)
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??(2 votes)
There are two numbers that you can square to get 25. 5^2 = 25, and (-5)^2 = 25. Thus, to solve the equation x^2 = 25, you have to account for both possibilities.(4 votes)
does anyone have an idea how the formula -b+-root b*-4ac/2a is derived ?(2 votes)- https://www.khanacademy.org/math/algebra/quadratics/quadratic_formula/v/proof-of-quadratic-formula
This video should be what you're looking for.(4 votes)
how would u solve this: (x+2)^2=10(0 votes)
There are two possible answers here, namely:Answer using the positive + square root:x=-2+√10Answer 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 sidestaking 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 squareroot.....
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!(12 votes)
8:45Video transcript
In this video, I'm going to do
several examples of quadratic equations that are really of a
special form, and it's really a bit of warm-up for the next
video that we're going to do on completing the square. So let me show you what
I'm talking about. So let's say I have 4x
plus 1 squared, minus 8 is equal to 0. Now, based on everything we've
done so far, you might be tempted to multiply this out,
then subtract 8 from the constant you get out here, and
then try to factor it. And then you're going to have
x minus something, times x minus something else
is equal to 0. And you're going to say, oh, one
of these must be equal to 0, so x could be that or that. We're not going to do that this
time, because you might see something interesting
here. We can solve this without
factoring it. And how do we do that? Well, what happens if
we add 8 to both sides of this equation? Then the left-hand side of the
equation becomes 4x plus 1 squared, and these
8's cancel out. The right-hand becomes
just a positive 8. Now, what can we do to both
sides of this equation? And this is just kind
of straight, vanilla equation-solving. This isn't any kind of
fancy factoring. We can take the square root of
both sides of this equation. We could take the square root. So 4x plus 1-- I'm just taking
the square root of both sides. You take the square root of both
sides, and, of course, you want to take the positive
and the negative square root, because 4x plus 1 could be the
positive square root of 8, or it could be the negative
square root of 8. So 4x plus 1 is equal to the
positive or negative square root of 8. Instead of 8, let me write
8 as 4 times 2. We all know that's what 8 is,
and obviously the square root of 4x plus 1 squared
is 4x plus 1. So we get 4x plus 1 is equal
to-- we can factor out the 4, or the square root of 4, which
is 2-- is equal to the plus or minus times 2 times the square
root of 2, right? Square root of 4 times square
root of 2 is the same thing as square root of 4 times the
square root of 2, plus or minus the square root of 4
is that 2 right there. Now, it might look like a really
bizarro equation, with this plus or minus 2 times
the square of 2, but it really isn't. These are actually two numbers
here, and we're actually simultaneously solving
two equations. We could write this as 4x plus
1 is equal to the positive 2, square root of 2, or 4x plus 1
is equal to negative 2 times the square root of 2. This one statement is equivalent
to this right here, because we have this plus or
minus here, this or statement. Let me solve all of these
simultaneously. So if I subtract 1 from
both sides of this equation, what do I have? On the left-hand side, I'm
just left with 4x. On the right-hand side, I
have-- you can't really mathematically, I mean, you
could do them if you had a calculator, but I'll just leave
it as negative 1 plus or minus the square root, or 2
times the square root of 2. That's what 4x is equal to. If we did it here, as two
separate equations, same idea. Subtract 1 from both sides of
this equation, you get 4x is equal to negative 1 plus 2,
times the square root of 2. This equation, subtract
1 from both sides. 4x is equal to negative
1 minus 2, times the square root of 2. This statement right here is
completely equivalent to these two statements. Now, last step, we just have to
divide both sides by 4, so you divide both sides by 4,
and you get x is equal to negative 1 plus or minus
2, times the square root of 2, over 4. Now, this statement is
completely equivalent to dividing each of these by 4,
and you get x is equal to negative 1 plus 2, times the
square root 2, over 4. This is one solution. And then the other solution is x
is equal to negative 1 minus 2 roots of 2, all
of that over 4. That statement and these two
statements are equivalent. And if you want, I encourage you
to-- let's substitute one of these back in, just so you
feel confident that something as bizarro as one of these
expressions can be a solution to a nice, vanilla-looking
equation like this. So let's substitute
it back in. 4 times x, or 4 times negative
1, plus 2 root 2, over 4, plus 1 squared, minus 8
is equal to 0. Now, these 4's cancel out, so
you're left with negative 1 plus 2 roots 2, plus
1, squared, minus 8 is equal to 0. This negative 1 and this
positive 1 cancel out, so you're left with 2 roots
of 2 squared, minus 8 is equal to 0. And then what are you
going to have here? So when you square this, you
get 4 times 2, minus 8 is equal to 0, which is true. 8 minus 8 is equal to 0. And if you try this one out,
you're going to get the exact same answer. Let's do another
one like this. And remember, these are special
forms where we have squares of binomials
in our expression. And we're going to see that the
entire quadratic formula is actually derived from a
notion like this, because you can actually turn any, you can
turn any, quadratic equation into a perfect square equalling
something else. We'll see that two
videos from now. But let's get a little
warmed up just seeing this type of thing. So let's say you have x squared
minus 10x, plus 25 is equal to 9. Now, once again your
temptation-- and it's not a bad temptation-- would be to
subtract 9 from both sides, so you get a 0 on the right-hand
side, but before you do that, just inspect this really fast.
And say, hey, is this just maybe a perfect square
of a binomial? And you see-- well, what two
numbers when I multiply them I get positive 25, and when I add
them I get negative 10? And hopefully negative
5 jumps out at you. So this expression right here is
x minus 5, times x minus 5. So this left-hand side can be
written as x minus 5 squared, and the right-hand
side is still 9. And I want to really
emphasize. I don't want this to ruin all of
the training you've gotten on factoring so far. We can only do this when this
is a perfect square. If you got, like, x minus 3,
times x plus 4, and that would be equal to 9, that would
be a dead end. You wouldn't be able to
really do anything constructive with that. Only because this is a perfect
square, can we now say x minus 5 squared is equal to 9, and
now we can take the square root of both sides. So we could say that x minus 5
is equal to plus or minus 3. Add 5 to both sides of this
equation, you get x is equal to 5 plus or minus 3, or x is
equal to-- what's 5 plus 3? Well, x could be 8 or x could be
equal to 5 minus 3, or x is equal to 2. Now, we could have done this
equation, this quadratic equation, the traditional
way, the way you were tempted to do it. What happens if you subtract
9 from both sides of this equation? You'll get x squared
minus 10x. And what's 25 minus 9? 25 minus 9 is 16, and that
would be equal to 0. And here, this would be just a
traditional factoring problem, the type that we've seen
in the last few videos. What two numbers, when you take
their product, you get positive 16, and when you sum
them you get negative 10? And maybe negative
8 and negative 2 jump into your brain. So we get x minus 8, times
x minus 2 is equal to 0. And so x could be equal to 8
or x could be equal to 2. That's the fun thing about
algebra: you can do things in two completely different ways,
but as long as you do them in algebraically-valid ways,
you're not going to get different answers. And on some level, if you
recognize this, this is easier because you didn't have to do
that little game in your head, in terms of, oh, what two
numbers, when you multiply them you get 16, and when you
add them you get negative 10? Here, you just said, OK, this is
x minus 5-- oh, I guess you did have to do it. You had to say, oh, 5 times 5
is 25, and negative 10 is negative 5 plus negative 5. So I take that back, you still
have to do that little game in your head. So let's do another one. Let's do one more of these, just
to really get ourselves nice and warmed up here. So, let's say we have x squared
plus 18x, plus 81 is equal to 1. So once again, we can
do it in two ways. We could subtract 1 from both
sides, or we could recognize that this is x plus
9, times x plus 9. This right here, 9 times 9
is 81, 9 plus 9 is 18. So we can write our equation
as x plus 9 squared is equal to 1. Take the square root of both
sides, you get x plus 9 is equal to plus or minus
the square root of 1, which is just 1. So x is equal to-- subtract 9
from both sides-- negative 9 plus or minus 1. And that means that x could be
equal to-- negative 9 plus 1 is negative 8, or x could be
equal to-- negative 9 minus 1, which is negative 10. And once again, you could have
done this the traditional way. You could have subtracted 1 from
both sides and you would have gotten x squared plus 18x,
plus 80 is equal to 0. And you'd say, hey, gee, 8 times
10 is 80, 8 plus 10 is 18, so you get x plus 8, times
x plus 10 is equal to 0. And then you'd get x could be
equal to negative 8, or x could be equal to negative 10. That was good warm up. Now, I think we're ready to
tackle completing the square.