Example 5: Using the quadratic formula | Quadratic equations | Algebra I | Khan Academy

Applying the Quadratic Formula

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We're asked to solve
the quadratic equation,

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

And it's already written
in standard form.

And there's many
ways to solve this.

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

So let me just rewrite it.

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

And actually, I'll
solve it twice

using the quadratic
formula to show you

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

the quadratic
formula will give us

the exact same roots or
the exact same solutions

to this equation.

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

Let's just remind ourselves
what the quadratic formula even

is actually.

That's a good place to start.

The quadratic formula
tells us that if we

have a quadratic
equation in the form ax

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

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

plus or minus the
square root of b

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

And this is derived from
completing the square

in a general way.

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

But this is the
quadratic formula.

This is actually giving
you two solutions,

because you have the
positive square root here

and the negative square root.

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

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

and c is equal to negative 3.

So applying the quadratic
formula right here,

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

b is 10.

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

of b squared.

b is 10.

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

So minus 4 times negative
3 times negative 3.

Let me just write it down.

Minus 4 times negative
3 times negative 3.

All of that's under
the radical sign.

And then all of that is over 2a.

So 2 times a is negative 6.

So this is going to be
equal to negative 10

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

negative 3 is positive 9.

Positive 9 times
4 is positive 36.

We have a minus sign out here.

So minus 36.

All of that over negative 6.

This is equal to
100 minus 36 is 64.

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

All of that over negative 6.

The principal square
root of 64 is 8.

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

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

So if we take the
positive version,

we say x could be
equal to-- negative 10

plus 8 is negative
2 over negative 6.

So that was taking
the plus version.

That's this right over here.

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

If we take the
negative square root,

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

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

is negative 18.

And that's going to
be over negative 6.

Negative 18 divided by
negative 6 is positive 3.

So the two roots for
this quadratic equation

are positive 1/3 and positive 3.

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

even if we manipulate this.

Some people might
not like the fact

that our first coefficient
here is a negative 3.

Maybe they want a positive 3.

So to get rid of
that negative 3,

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

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

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

which is still equal to 0.

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

and c is equal to 3 again.

And we could apply
the quadratic formula.

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

So negative negative
10 is positive

10, plus or minus
the square root

of b squared, which is
negative 10 squared,

which is 100, minus
4 times a times c.

a times c is 9 times 4 is 36.

So minus 36.

All of that over 2 times a.

All of that over 6.

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

or really that's
just going to be 8.

All of that over 6.

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

We get x could be equal to 3.

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

here, 10 minus 8 is 2.

2 over 6 is 1/3.

So once again, you get
the exact same solutions.