Example 1: Using the quadratic formula

Algebra

Quadratics

The quadratic formula (quadratic equation)

How to Use the Quadratic Formula
Example: Quadratics in standard form
Proof of Quadratic Formula
Example 1: Using the quadratic formula
Example 2: Using the quadratic formula
Example 3: Using the quadratic formula
Example 4: Applying the quadratic formula
Example 5: Using the quadratic formula
Quadratic formula
Example: Complex roots for a quadratic
Discriminant of Quadratic Equations
Discriminant for Types of Solutions for a Quadratic
Solutions to quadratic equations

Example 1: Using the quadratic formula Quadratic Formula 1

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Example 1: Using the quadratic formula

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Use the quadratic formula to solve the equation, 0 is equal
to negative 7q squared plus 2q plus 9.
Now, the quadratic formula, it applies to any quadratic
equation of the form-- we could put the 0 on
the left hand side.
0 is equal to ax squared plus bx plus c.
And we generally deal with x's, in this problem we're
dealing with q's.
But the quadratic formula says, look, if you have a
quadratic equation of this form, that the solutions of
this equation are going to be x is going to be equal to
negative b plus or minus the square root of b squared minus
4ac-- all of that over 2a.
And this is actually two solutions here, because
there's one solution where you take the positive square root
and there's another solution where you take
the negative root.
So it gives you both roots of this.
So if we look at the quadratic equation that we need to solve
here, we can just pattern match.
We're dealing with q's, not x's, but this is the same
general idea.
It could be x's if you like.
And if we look at it, negative 7 corresponds to a.
That is our a.
It's the coefficient on the second degree term.
2 corresponds to b.
It is the coefficient on the first degree term.
And then 9 corresponds to c.
It's the constant.
So, let's just apply the quadratic formula.
The quadratic formula will tell us that the solutions--
the q's that satisfy this equation-- q will be equal to
negative b.
b is 2.
Plus or minus the square root of b squared, of 2 squared,
minus 4 times a times negative 7 times c, which is 9.
And all of that over 2a.
All of that over 2 times a, which is once
again negative 7.
And then we just have to evaluate this.
So this is going to be equal to negative 2 plus or minus
the square root of-- let's see, 2 squared is 4-- and then
if we just take this part right here, if we just take
the negative 4 times negative 7 times 9, this negative and
that negative is going to cancel out.
So it's just going to become a positive number.
And 4 times 7 times 9.
4 times 9 is 36.
36 times 7.
Let's do it up here.
36 times 7.
7 times 6 is 42.
7 times 3, or 3 times 7 is 21.
Plus 4 is 25.
252.
So this becomes 4 plus 252.
Remember, you have a negative 7 and you
have a minus out front.
Those cancel out, that's why we have a positive 252 for
that part right there.
And then our denominator, 2 times negative
7 is negative 14.
Now what does this equal?
Well, we have this is equal to negative 2 plus or minus the
square root of-- what's 4 plus 252?
It's just 256.
All of that over negative 14.
And what's 256?
What's the square root of 256?
It's 16.
You can try it out for yourself.
This is 16 times 16.
So the square root of 256 is 16.
So we can rewrite this whole thing as being equal to
negative 2 plus 16 over negative 14.
Or negative 2 minus-- right?
This is plus 16 over negative 14.
Or minus 16 over negative 14.
If you think of it as plus or minus, that plus is that plus
right there.
And if you have that minus, that minus is
that minus right there.
Now we just have to evaluate these two numbers.
Negative 2 plus 16 is 14 divided by negative 14 is
negative 1.
So q could be equal to negative 1.
Or negative 2 minus 16 is negative 18 divided by
negative 14 is equal to 18 over 14.
The negatives cancel out, which is equal to 9 over 7.
So q could be equal to negative 1, or it could be
equal to 9 over 7.
And you could try these out, substitute these q's back into
this original equation, and verify for yourself that they
satisfy it.
We could even do it with the first one.
So if you take q is equal to negative 1.
Negative 7 times negative 1 squared-- negative 1 squared
is just 1-- so this would be negative 7 times 1, right?
That's negative 1 squared.
Negative 1 times 2 is minus 2 plus 9.
So it's negative 7 minus 2, which is negative 9, plus 9,
does indeed equal 0.
So this checks out.
And I'll leave it up to you to verify that 9 over
7 also works out.

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