WEBVTT
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Let's take a look at solving quadratic equations.
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Quadratic equations are degree-two equations.
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You'll notice the highest exponent on the
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variables is two, which means it's quadratic.
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You will need to be able to solve these all throughout the semester.
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Let's look at this one: 4 x squared plus 7 x equals 2.
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This one is not in standard form, which means
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standard form is when you have descending powers
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and zero on the right hand side.
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So, to get this in standard form we're going to
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first subtract 2 from both sides of the equation.
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Now it's in standard form.
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We want to see if this equation will solve by
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factoring, which is one of the techniques that
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we'll use a lot to solve quadratic equations.
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Not all quadratics factor, but this one does.
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Let's look at it.
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To solve, to factor,
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you're going to have two expressions.
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Now, a couple of good notes: I see 4 x squared
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here, and I have choices.
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I can put 2 x and 2 x, but if I do that, the only
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factors of 2 are 2 and 1.
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And so then, that means I'd have a 2 x and a 2 in
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one of these factors, which means there'd be a
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common factor of 2.
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Notice there's not a common factor 2 to begin
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with, so I know that, by trial and error, it's
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not going to be 2 x and 2 x.
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I'm going to try 4 x and x, and then
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I need 2 and 1.
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Well, I want to get a 7 x in the middle, so I want
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to get 8 minus 1 to give me the 7 x in the middle
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with a plus 8 x and a minus x, so it does factor.
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Once I get it factored, I simply set
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each factor equal to zero.
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So, I'm going to have 4 x minus 1 equals 0 or x
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plus 2 equals 0, and I'll solve each of these equations.
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Four x equals 1, so one of my solutions is x
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equals 1/4, and the other solution is x equals
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negative 2, so I end up with two solutions here
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to this quadratic equation.
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So this is an example of solving by factoring.
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Factoring is a very important skill that you'll
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need to be able to do throughout the semester, so
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you need to practice that and make sure that you can solve by factoring.
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Alright, next we're going to look at solving quadratic equations by other techniques.
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Let's look at another a quadratic equation.
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This one is quantity 3 x minus 2 squared equals 16.
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If we expand this out, the first term would be 9 x
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squared, so we notice it's quadratic.
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But importantly, whenever you see a quadratic in
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this form of something squared equals a number,
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there is a technique or strategy called the
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square root method that works really nice.
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If you think about this, you're looking for some
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number that squared gives you 16.
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Well of course, there's two numbers that you could square and get 16.
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There's positive 4 and negative 4.
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So the way we write this using the square root
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method is 3 x minus 2 equals plus or minus 4.
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Again, positive 4 squared gives me 16 and
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negative 4 squared gives me 16, so there's always
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a plus or minus when using the square root method.
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Now, this actually means two separate equations:
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3 x minus 2 equals 4 or 3 x minus 2 equals a
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negative 4, and then we'll solve each of those equations.
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We're going to get on the left 3x equals 6, so x
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equals 2. From the right hand side, you're
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going to have 3 x equals negative 2, so x equals negative 2/3.
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So we end up with these two solutions, and what
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you want to remember is whenever your quadratic
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is in this form or can easily be put in this form,
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something squared equals a number, you want
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to think about using the square root method, and
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you want to remember not to forget the plus or
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minus because you can either square a positive
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number and get 16 or you could square a
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negative number and get 16.
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So there's always a plus or minus that you need to remember.
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That's a common thing students forget to put.
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Alright, let's look at another quadratic equation.
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Let's look at another quadratic equation, this
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one: 2 x, parentheses x minus 1, equals 5.
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Notice that if we expand it out, we have 2 x
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squared, so it's a quadratic equation.
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So first, I'm going to multiply on the left hand
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side and get 2 x squared minus 2 x, going to go and subtract the 5,
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get this quadratic in standard form.
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Now it's in standard form, but it doesn't factor
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easily, so we have another technique to solve
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this quadratic called the quadratic formula.
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First, it has to be in standard form, which this one is.
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Standard form is when it looks like a x squared
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plus b x plus c equals 0. The quadratic
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formula is x equals negative b plus or minus the
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square root of b squared minus 4 a c,
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all of this is over 2 a.
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This is a very important formula that you need to remember.
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A lot of students forget that the b is over 2a,
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the whole numerator is over 2a, so you need to remember this formula.
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Notice that we have our quadratic in standard
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form: our a is 2, our b is negative 2, and our c is negative 5.
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So I'll use this formula, plugging in my numbers.
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I'm going to get negative b, which will be a
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positive 2, plus or minus the square root. When
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you square b, you have negative 2 squared minus
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4 a c, and all of this is over 2 a.
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Okay, so now we're going to simplify a little bit more.
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Underneath the square root, what are we going to have?
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Let's see, we're going to have negative 2
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squared, which is of course 4, and then notice
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that we have two negative signs so that's going
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to change to plus, so we'll have 4 plus 40, which
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is 44, under the square root. Then in the denominator, we have 4.
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Okay, so we've gotten here to 2 plus or minus the
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square root of 44 over 4. Now comes the
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tricky part for students, simplifying this.
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First of all, can we simplify the square root of 44?
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Are there any perfect squares in 44?
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We need to look for the largest one, which in
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this case is 4, so I can first reduce the square root of 44.
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It'll be 4 times 11; 4 is the biggest one. So I
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write this as 2 square root of 11 over 4, and now
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we need to reduce this further.
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There's lots of places that students make mistakes.
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One of the mistakes they make over here is
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they're so tempted to just put 1/2 and forget
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about the 4 that goes underneath this term in the
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numerator as well, so don't do that. Don't write
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1/2 plus the square root of 44.
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Now when we get to here, we look at these three
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terms, and we look for the greatest common factor
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in those three terms, which of course is a 2.
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So we're going to reduce each term by a factor of
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2. So 2 divided by 2 is 1, and when we do 2 into
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this we have the square root of 11,
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and 2 into 4 goes 2.
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So, our final answer is 1 plus or minus the
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square root of 11 over 2. You need to be very
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careful on problems like this where you reduce.
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A lot of students will just forget about this 1.
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They'll just write plus or minus the square root
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of 11 over 2 and think that 2 goes into 2 maybe 0
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times. But remember there should be two terms in the answer
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if there was two terms up here, then there must be two here.
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So this, of course, can also be written 1 plus
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the square root of 11 over 2 and then 1 minus the square root of 11 over 2.
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Be very careful when reducing on these types
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of quadratics. You want to practice these a lot so that you get this.
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This is one of the problems that students missed a lot on one of the tests.
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Alright, let's look at another quadratic equation.
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Let's look at this example here: 4 plus 1 over x minus 1 over x squared.
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Notice it's another rational equation, remember I
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said rational equations you have fractions.
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So we've got variables in the denominator.
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First thing we want to do is to clear fractions.
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We want to multiply by the least common
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denominator, which will be x squared.
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So if we do that, we'll get 4 x squared plus x minus 1 equals 0.
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So notice that it's now a quadratic equation, and
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we want to solve it, and again it doesn't factor
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easily, so we'll again use the quadratic formula.
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Do you remember the quadratic formula?
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Think about it for a minute and write it down.
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It's in standard form, so we're going to use the quadratic formula.
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And we're going to get negative b, b is 1, so we
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get negative 1 plus or minus the square root of b
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squared is 1 minus 4 times a, which is 4, times c which is a negative 1,
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and all of this is over 2 a.
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So it's very important that you remember the quadratic formula.
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Let's see what we get under the radical.
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We have one and again there's two negative signs
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so it's plus 16, 1 plus 16 is 17, and in the denominator we get 8.
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Now this particular answer doesn't simplify, the
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square root of 17 can't be simplified,
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so we are finished now.
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Our solutions are negative 1 plus the square root
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of 17 over 8, and the other solution is negative
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1 minus the square root of 17 over 8.
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So again, think about the different quadratics,
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think about which method works best.
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Sometimes factoring works great, sometimes you
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want to use the square root method, and sometimes
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you want to use the quadratic formula.
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All of those techniques are very important.
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First thing, remember you need to identify and recognize equations.
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This is quadratic because it's degree-two, and
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think there's three different methods we talked
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about: factoring, square root method, and the
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quadratic formula. You need to practice on
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all of these until you are successful with each technique.