WEBVTT

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We're asked to solve
the quadratic equation,

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negative 3x squared plus
10x minus 3 is equal to 0.

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And it's already written
in standard form.

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And there's many
ways to solve this.

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But in particular, all solve
it using the quadratic formula.

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So let me just rewrite it.

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We have negative 3x squared
plus 10x minus 3 is equal to 0.

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And actually, I'll
solve it twice

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using the quadratic
formula to show you

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that as long as we manipulated
this in the valid way,

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the quadratic
formula will give us

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the exact same roots or
the exact same solutions

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to this equation.

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So in this form right over
here, what are our ABCs?

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Let's just remind ourselves
what the quadratic formula even

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is actually.

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That's a good place to start.

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The quadratic formula
tells us that if we

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have a quadratic
equation in the form ax

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squared plus bx plus c is equal
to 0, so in standard form,

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then the roots of this are
x are equal to negative b

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plus or minus the
square root of b

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squared minus 4ac,
all of that over 2a.

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And this is derived from
completing the square

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in a general way.

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So it's no magic here, and I've
derived it in other videos.

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But this is the
quadratic formula.

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This is actually giving
you two solutions,

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because you have the
positive square root here

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and the negative square root.

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So let's apply it here in the
case where-- in this case,

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a is equal to negative
3, b is equal to 10,

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and c is equal to negative 3.

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So applying the quadratic
formula right here,

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we get our solutions to be
x is equal to negative b.

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b is 10.

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So negative b is negative 10
plus or minus the square root

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of b squared.

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b is 10.

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So b squared is 100
minus 4 times a times c.

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So minus 4 times negative
3 times negative 3.

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Let me just write it down.

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Minus 4 times negative
3 times negative 3.

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All of that's under
the radical sign.

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And then all of that is over 2a.

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So 2 times a is negative 6.

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So this is going to be
equal to negative 10

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plus or minus the square root
of 100 minus-- negative 3 times

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negative 3 is positive 9.

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Positive 9 times
4 is positive 36.

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We have a minus sign out here.

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So minus 36.

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All of that over negative 6.

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This is equal to
100 minus 36 is 64.

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So negative 10 plus or
minus the square root of 64.

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All of that over negative 6.

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The principal square
root of 64 is 8.

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But we're taking the positive
and negative square root.

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So this is negative 10 plus
or minus 8 over negative 6.

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So if we take the
positive version,

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we say x could be
equal to-- negative 10

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plus 8 is negative
2 over negative 6.

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So that was taking
the plus version.

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That's this right over here.

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And negative 2 over
negative 6 is equal to 1/3.

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If we take the
negative square root,

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negative 10 minus 8-- So let's
take negative 10 minus 8.

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That would be x is equal
to-- negative 10 minus 8

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is negative 18.

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And that's going to
be over negative 6.

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Negative 18 divided by
negative 6 is positive 3.

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So the two roots for
this quadratic equation

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are positive 1/3 and positive 3.

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And I want to show you the
we'll get the same answer,

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even if we manipulate this.

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Some people might
not like the fact

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that our first coefficient
here is a negative 3.

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Maybe they want a positive 3.

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So to get rid of
that negative 3,

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they can multiply both sides of
this equation times negative 1.

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And then if you did
that, you would get 3x

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squared minus 10x plus 3 is
equal to 0 times negative 1,

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which is still equal to 0.

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So in this case, a is equal to
3, b is equal to negative 10,

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and c is equal to 3 again.

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And we could apply
the quadratic formula.

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We get x is equal to
negative b. b is negative 10.

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So negative negative
10 is positive

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10, plus or minus
the square root

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of b squared, which is
negative 10 squared,

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which is 100, minus
4 times a times c.

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a times c is 9 times 4 is 36.

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So minus 36.

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All of that over 2 times a.

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All of that over 6.

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So this is equal to 10 plus or
minus the square root of 64,

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or really that's
just going to be 8.

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All of that over 6.

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If we add 8 here, we get
10 plus 8 is 18 over 6.

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We get x could be equal to 3.

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Or if we take the negative
square root or the negative 8

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here, 10 minus 8 is 2.

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2 over 6 is 1/3.

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So once again, you get
the exact same solutions.