Methods For Solving Quadratic Equations Worksheet With Answers Page 2
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1
23. COMPLETING THE SQUARE
2
If the quadratic equation is of the form
ax
bx
c
, 0
where
a
0
and the quadratic expression is
not factorable, try completing the square.
x
2
Example:
x
6
11
0
, divide all terms by “a” before proceeding to the next steps.
**Important: If
a
1
x
2
Move the constant to the right side
x
6
_____
11
_____
and supply a blank on each side
b
6
Find half of b, which means
:
3
2
2
2
b
2
Square half of b:
:
3
9
2
2
b
x
2
Add
to both sides of the equation
x
6
9
11
9
2
Factor the quadratic side
(
x
3
)(
x
) 3
20
(which is a perfect square because you just made it that way!)
2
Then write in perfect square form
(
x
) 3
20
2
Take the square root of both sides
(
x
) 3
20
3
x
20
x
3
20
Simplify
the
radical
Solve for x
x
3
2
5
This represents the exact answer.
Decimal approximations can be found using a
calculator: 1.472 and -7.472
4. QUADRATIC FORMULA
2
Any quadratic equation of the form
ax
bx
c
, 0
where
a
0
can be solved for both real and
imaginary solutions using the quadratic formula:
2
b
b
4
ac
x
2
a
2
x
6
x
11
0
(
a
, 1
b
, 6
c
11
)
Example:
Substitute values into the quadratic formula:
2
6
6
4
1 (
)(
11
)
6
36
44
6
80
x
x
x
simplify
the
radical
2
) 1 (
2
2
6
4
5
x
x
3
2
5
This
is
the
final
simplified
EXACT
answer
2
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