The quadratic formula (quadratic equation)
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How to Use the Quadratic Formula
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Example: Quadratics in standard form
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Proof of Quadratic Formula
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Example 1: Using the quadratic formula
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Example 2: Using the quadratic formula
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Example 3: Using the quadratic formula
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Example 4: Applying the quadratic formula
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Example 5: Using the quadratic formula
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Quadratic formula
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Example: Complex roots for a quadratic
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Discriminant of Quadratic Equations
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Discriminant for Types of Solutions for a Quadratic
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Solutions to quadratic equations
Example 2: Using the quadratic formula Quadratic Formula 2
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- Use the quadratic formula to solve the equation, negative x
- squared plus 8x is equal to 1.
- Now, in order to really use the quadratic equation, or to
- figure out what our a's, b's and c's are, we have to have
- our equation in the form, ax squared plus bx plus c is
- equal to 0.
- And then, if we know our a's, b's, and c's, we will say that
- the solutions to this equation are x is equal to negative b
- plus or minus the square root of b squared minus 4ac-- all
- of that over 2a.
- So the first thing we have to do for this equation right
- here is to put it in this form.
- And on one side of this equation, we have a negative x
- squared plus 8x, so that looks like the first two terms. But
- our constant is on the other side.
- So let's get the constant on the left hand side and get a 0
- here on the right hand side.
- So let's subtract 1 from both sides of this equation.
- The left hand side of the equation will become negative
- x squared plus 8x minus 1.
- And then the right hand side, 1 minus 1 is 0.
- Now we have it in that form.
- We have ax squared a is negative 1.
- So let me write this down. a is equal to negative 1.
- a is equal to negative 1.
- It's implicit there, you could put a 1 here if you like.
- A negative 1.
- Negative x squared is the same thing as negative 1x squared.
- b is equal to 8.
- So b is equal to 8, that's the 8 right there.
- And c is equal to negative 1.
- That's the negative 1 right there.
- So now we can just apply the quadratic formula.
- The solutions to this equation are x is equal to negative b.
- Plus or minus the square root of b squared, of 8 squared,
- minus 4ac-- let me do it in that green color --minus 4,
- the green is the part of the formula.
- The colored parts are the things that we're substituting
- into the formula.
- Minus 4 times a, which is negative 1, times negative 1,
- times c, which is also negative 1.
- And then all of that-- let me extend the square root sign a
- little bit further --all of that is going to be
- over 2 times a.
- In this case a is negative 1.
- So let's simplify this.
- So this becomes negative 8, this is negative 8, plus or
- minus the square root of 8 squared is 64.
- And then you have a negative 1 times a negative 1, these just
- cancel out just to be a 1.
- So it's 64 minus is 4.
- That's just that 4 over there.
- All of that over negative 2.
- So this is equal to negative 8 plus or minus the
- square root of 60.
- All of that over negative 2.
- And let's see if we can simplify the radical
- expression here, the square root of 60.
- Let's see, 60 is equal to 2 times 30.
- 30 is equal to 2 times 15.
- And then 15 is 3 times 5.
- So we do have a perfect square here.
- We do have a 2 times 2 in there.
- It is 2 times 2 times 15, or 4 times 15.
- So we could write, the square root of 60 is equal to the
- square root of 4 times the square root of 15, right?
- The square root of 4 times the square root of 15,
- that's what 60 is.
- 4 times 15.
- And so this is equal to-- square root of 4 is 2 times
- the square of 15.
- So we can rewrite this expression, right here, as
- being equal to negative 8 plus or minus 2 times the square
- root of 15, all of that over negative 2.
- Now both of these terms right here are divisible by either 2
- or negative 2.
- So let's divide it.
- So we have negative 8 divided by negative 2, which is
- positive 4.
- So let me write it over here.
- Negative 8 divided by negative 2 is positive 4.
- And then you have this weird thing.
- Plus or minus 2 divided by negative 2.
- And really what we have here is 2 expressions.
- But if we're plus 2 and we divide by negative 2, it will
- be negative 1.
- And if we take negative 2 and divide by negative 2, we're
- going to have positive 1.
- So instead of plus or minus, you could imagine it is going
- to be minus or plus.
- But it's really the same thing.
- Right?
- It's really now minus or plus.
- If it was plus, it's now going to be a minus.
- If it was a minus, it's now going to be a plus.
- Minus or plus 2 times the square root of 15.
- Or another way to view it is that the two solutions here
- are 4 minus two roots of 15, and 4 plus two roots of 15.
- These are both values of x that'll satisfy this equation.
- And if this confuses you, what I did, turning a plus or minus
- into minus plus.
- Let me just take a little bit of an aside there.
- I could write this expression up here as two expressions.
- That's what the plus or minus really is.
- There's a negative 8 plus 2 roots of 15 over negative 2.
- And then there's a negative 8 minus 2 roots of 15 over
- negative 2.
- This one simplifies to-- negative 8 divided by
- negative 2 is 4.
- 2 divided by negative 2 is negative 1.
- 2 times a 4 minus the square root of 15.
- And then over here you have negative 8 divided by negative
- 2, which is 4.
- And then negative 2 divided by negative 2, which is plus the
- square of 15.
- And I just realized I made a mistake up here.
- When we're dividing a 2 divided by negative 2, we
- don't have this 2 over here.
- This is just a plus or minus the root of 15.
- We just saw that when I did it out here.
- So this is minus the square root of 15.
- And this is plus the square root of 15.
- So the two solutions for this equation-- It's good that I
- took that little hiatus there, that little aside there.
- The two solutions could be 4 minus the square root of 15,
- or x, or and, x could be 4 plus the square root of 15.
- Either of those values of x will satisfy this original
- quadratic equation.
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