Quadratic odds and ends
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Solving a quadratic by factoring
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CA Algebra I: Factoring Quadratics
-
Algebra II: Quadratics and Shifts
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Examples: Graphing and interpreting quadratics
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CA Algebra I: Completing the Square
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Introduction to the quadratic equation
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Quadratic Equation part 2
-
Quadratic Formula (proof)
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CA Algebra I: Quadratic Equation
-
CA Algebra I: Quadratic Roots
Quadratic Equation part 2 2 more examples of solving equations using the quadratic equation
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- Welcome to part two of the presentation on
- quadratic equations.
- Well, I think I thoroughly confused you the last time
- around, so let me see if I can fix that a bit by doing
- several more examples.
- Entonces comencemos con una revision de lo que
- So let's just start with a review of what the
- quadratic equation is.
- The quadratic equation says, if I'm trying to solve the
- equation Ax squared plus Bx plus C equals 0, then the
- solution or the solutions because there's usually two
- times that it intersects the x-axis, or two solutions for
- this equation is x equals minus B plus or minus the square root
- of B squared minus 4 times A times C.
- And all of that over 2A.
- So let's do a problem and hopefully this should make
- a little more sense.
- That's a 2 on the bottom.
- So let's say I had the equation minus 9x squared minus
- 9x plus 6 equals 0.
- So in this example what's A?
- Well, A is the coefficient on the x squared term.
- The x squared term is here, the coefficient is minus 9.
- So let's write that.
- A equals minus 9.
- What does B equal?
- B is the coefficient on the x term, so that's this term here.
- So B is also equal to minus 9.
- And C is the constant term, which in this example is 6.
- So C is equal to 6.
- Now we just substitute these values into the actual
- quadratic equation.
- So negative B, so it's negative times negative 9.
- That's B.
- Plus or minus the square root of B squared, so that's 81.
- Right?
- Negative 9 squared.
- Minus 4 times negative 9.
- That's A.
- Times C, which is 6.
- And all of that over 2 times negative 9, which
- is minus 18, right?
- 2 times negative 9-- 2A.
- Let's try to simplify this up here.
- Well, negative negative 9, that's positive 9.
- Plus or minus the square root of 81.
- Let's see.
- This is negative 4 times negative 9.
- Negative 4 times negative 9 is positive 36.
- And then positive 36 times 6 is-- let's see.
- 30 times 6 is 180.
- And then 180 plus another 36 is 216.
- Plus 216, is that right?
- 180 plus 36 is 216.
- All of that over 2A.
- 2A we already said is minus 19.
- So we simplify that more.
- That's 9 plus or minus the square root 81 plus 216.
- That's 80 plus 217.
- That's 297.
- And all of that over minus 18.
- Now, this is actually-- the hardest part with the quadratic
- equation is oftentimes just simplifying this expression.
- We have to figure out if we can simplify this radical.
- Well, let's see.
- One way to figure out if a number is divisible by 9 is to
- actually add up the digits and see if the digits
- are divisible by 9.
- In this case, it is.
- 2 plus 9 plus 7 is equal to 18.
- So let's see how many times 9 goes into that.
- I'll do it on the side here; I don't want to be too messy.
- 9 goes into 2 97.
- 3 times 27.
- 27-- it goes 33 times, right?
- So this is the same thing as 9 plus or minus the square root
- of 9 times 33 over minus 18.
- And 9 is a perfect square.
- That's why I actually wanted to see if 9 would work because
- that's the only way I could get it out of the radical, if
- it's a perfect square.
- As you learned in that exponent rules number one module.
- So this is equal to 9 plus or minus 3 times the square
- root of 33, and all of that over minus 18.
- We're almost done.
- We can actually simplify it because 9, 3, and minus 18
- are all divisible by 3.
- Let's divide everything by 3.
- 3 plus or minus the square root of 33 over minus 6.
- And we are done.
- So as you see, the hardest thing with the quadratic
- equation is often just simplifying the expression.
- But what we've said, I know you might have lost track-- we did
- all this math-- is we said, this equation: minus 9x
- squared minus 9x plus 6.
- Now we found two x values that would satisfy this equation
- and make it equal to 0.
- One x value is x equals 3 plus the square root
- of 33 over minus 6.
- And the second value is 3 minus the square root
- of 33 over minus 6.
- And you might want to think about why we have
- that plus or minus.
- We have that plus or minus because a square root could
- actually be a positive or a negative number.
- Let's do another problem.
- Hopefully this one will be a little bit simpler.
- Let's say I wanted to solve minus 8x squared
- plus 5x plus 9.
- Now I'm going to assume that you've memorized the quadratic
- equation because that's something you should do.
- Or you should write it down on a piece of paper.
- But the quadratic equation is negative B-- So b is 5, right?
- We're trying to solve that equal to 0, so negative B.
- So negative 5, plus or minus the square root of B squared-
- that's 5 squared, 25.
- Minus 4 times A, which is minus 8.
- Times C, which is 9.
- And all of that over 2 times A.
- Well, A is minus 8, so all of that is over minus 16.
- So let's simplify this expression up here.
- Well, that's equal to minus 5 plus or minus
- the square root of 25.
- Let's see.
- 4 times 8 is 32 and the negatives cancel out, so
- that's positive 32 times 9.
- Positive 32 times 9, let's see.
- 30 times 9 is 270.
- It's 288.
- I think.
- Right?
- 288.
- We have all of that over minus 16.
- Now simplify it more.
- Minus 5 plus or minus the square root-- 25 plus
- 288 is 313 I believe.
- And all of that over minus 16.
- And I think, I'm not 100% sure, although I'm pretty sure.
- I haven't checked it.
- That 313 can't be factored into a product of a perfect
- square and another number.
- In fact, it actually might be a prime number.
- That's something that you might want to check out.
- So if that is the case and we've got it in completely
- simplified form, and we say there are two solutions, two
- x values that will make this equation true.
- One of them is x is equal to minus 5 plus the square
- root of 313 over minus 16.
- And the other one is x is equal to minus 5 minus the square
- root of 313 over minus 16.
- Hopefully those two examples will give you a good
- sense of how to use the quadratic equation.
- I might add some more modules.
- And then, once you master this, I'll actually teach you how to
- solve quadratic equations when you actually get a negative
- number under the radical.
- Very interesting.
- Anyway, I hope you can do the module now and maybe I'll add a
- few more presentations because this isn't the easiest module.
- But I hope you have fun.
- Bye.
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