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Quadratic Formula: Example 1
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Let's apply these concepts:
Standard Form
Quadratic Formula
Discriminant
as we solve
In order to use the quadratic formula, what do we need to do first?
We need to rewrite the equation in
Standard Form
.
Express the Standard Form of a Quadratic equation.
Must the order of the powers be like this?
Yes, A must be the coefficient of
, etc.
Rewrite the problem in the Standard Form.
List the values of
A
,
B
, and
C
.
A
= 3,
B
= – 12, and
C
= 2
In general, must the equation be rewritten until the coefficient of
x
2
is positive?
No.
How much rewriting must we do?
Enough to get all of the terms on one side of the equal sign and in order of descending powers, as above.
Now we are ready to consider the
Quadratic Formula
.
State it in terms of
One of the terms in the Quadratic Formula is –
B
.
In this problem, will we have – 12 or – (– 12) for this term?
–
B
= – (– 12) = +12.
Under the radical sign, we see
B
2
. In this problem, will that become 144 or – 144?
Since that term is
B
2
= (– 12)
2
, we get +144.
Will that term always be positive?
Yes.
In the quadratic formula, we have a denominator. Is it divided into "everything"?
Yes, so it is important to draw the "line" under the whole numerator.
Is
under the square root sign?
Yes, so it is important to draw the top of the radical long enough to include both terms.
Now let's substitute our
A
= 3,
B
= – 12, and
C
= 2
into the quadratic formula.
Simplify the terms under the radical.
Now use these results to solve for
x
.
Can we divide out the 2's? Why?
Yes, because they are a common factor.
Do it.
Can we divide the 3?
No, because we can factor only a
out of the
.
is the exact solution.
Find decimal approximations to 2 decimal places.
x
~ 3.83,
x
~ 0.17
Let’s check the exact solution. Substitute it into
the Standard Form of this quadratic equation.
Square the first term.
Remove the parentheses.
Note the reversal of the second ± sign.
Combine the like terms.
What happens to the radical terms?
They eliminate.
Combine them.
We get 22 – 24 + 2 ? 0, or
0 = 0, which checks.
Another concept to apply here is the
Discriminant,
Write the general expression for the discriminant
in terms of
A
,
B
,
C
.
Substitute the values from this example:
Do the math.
Is this consistent with the two roots, which we found?
Yes.
Why?
When the discriminant is positive, there are two distinct real roots.
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General Contents
Detailed Contents
Index
