We need to rewrite the equation in Standard Form.
Do it.
Ax2 + Bx + C = 3x2 -12x +2 = 0
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 x2 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.
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 B2. In this problem, will that become 144 or -144?
Since that term is B2 = (-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 -4AC 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
Check the exact solution.
, or
, or
, or
, or
22 - 24 + 2 ? 0, or
0 = 0
Example 2: Apply the quadratic formula to solve
x2 - x - 12 = 0
List the values of A, B, and C.
A = 1, B = -1, C = -12
Substitute these values in the quadratic formula.
Simplify the radical.
Us this result to solve for x.
Find the solution which corresponds to choosing +7.
Find the solution which corresponds to choosing -7.
Check x = 4 in the original equation.
For x = 4 in x2 - x - 12 = 0, we get
42 - 4-12 = 16 - 16 = 0 OK
Check x = -3 in the original equation.
For x = -3 in x2 - x - 12 = 0, we get
(-3)2 - (-3)-12 = 9 + 3 - 12 = 0 OK
As a review, try factoring x2 - x - 12 = 0
We get (x - 4)*(x + 3) = 0
Apply the zero product rule.
We get
x - 4 = 0 and x + 3 = 0, or
x = 4 and x = -3, just as with the quadratic formula.
As further review, let's try Completing-the-Square on this same problem: x2 - x - 12 = 0.
What do we do first?
Move the constant term to the other side.
Do it.
x2 - x = 12
What do we need to add to each side in order to make the left side a perfect square?
Do it.
Do the "squares".
Combine the constants on the right-hand side and write the left side as a perfect square.
Take the square root of both sides.
Solve for x.
, as before.
We need to rewrite the problem in Standard Form.
Do the multiplication.
3x2 + 2x - 9x - 6 = x2 - 2
Collect like terms on the left side.
3x2 - 7x - 6 = x2 - 2
Subtract (x2 - 2) from both sides.
3x2 - 7x - 6 - (x2 - 2) = 0
Collect like terms again.
2x2 - 7x - 4 = 0
Is this in standard form?
Yes.
List the values of A, B, C.
A = 2, B = -7, C = -4
Substitute in the quadratic formula.
Simplify the square root.
Use this result to solve for x.
Check 4 in the original equation,
(x - 3)*(3x + 2) = x2 - 2.
We get
(4 - 3)*(3*4+2) ? 42 - 2, or
1*(12+2) ? 16 - 2, or
14 = 14 OK.
Check 
We get
, or
, or
, or
OK
B2 - 4AC
Substitute the values from example 2:
A = 2, B = -7, C = -4
(-7)2 - 4(-2)*(-4)
Do the math.
49 -(-32) = 49 + 32 = 81
Is this consistent with the two roots,
x = 4, and x = -1/2, which we found?
Yes.
Why?
When the discriminant is positive, there are two distinct real roots.
Example 4: Solve x2 - 3x + 5 = 0.
First evaluate the discriminant.
B2 - 4AC = (-3)2 - 4(1)(5) = 9 - 20 = -11
With this result, what should we expect about the roots?
We expect two complex roots.
Apply the quadratic formula.
Check this result.
, or
The square root terms subtract out and we get
, or
-5 + 5 = 0 OK