2.6 Solving Quadratic Equations
with Complex Roots
11/9/2012
Completing the square
Solve x 2 – 2x + 3 = 0 by completing the square.
x 2 – 2x + 3 = 0
(x 2 – 2x +1) + 3 – 1 = 0
(x – 1) 2 + 2 = 0
( x – 1)2 = – 2 Subtract 2 from both sides.
x– 1= +
– –2
Take the square root of each side.
x=1 +
– – 2 Add 1 to each side.
−2 = −1 • 2
ANSWER
 = −1
x=1+
– i 2 Write in terms of i.
The solutions are 1 + i 2 and 1 – i 2 .
Sum of Squares pattern
Find complex solution of x2 + 49 = 0
x 2 + 49 = 0
- 49 - 49
x2 = -49
x = -1 • 49
x = ± 7i
Sum of Squares pattern
Find complex solution of 25x2 + 9 = 0
25x 2 + 9 = 0
-9 -9
25x2 = -9
25
25
9
2
x = 25
x = −1 •
x=±
3
i
5
9
25
Quadratic
Formula:
Is used to solve quadratic equations written
in the form ax2 + bx +c = 0
b
b  4 ac
2
2a
Solve an Equation with Imaginary Solutions
–b +
b 2 – 4ac
–
x =
2a
Solve x 2 + 2x + 2 = 0.
SOLUTION
(
–2 +
22 – 4 (1 ( 2
–
x =
2 (1
(
(
–4
–2 +
–
x =
2
2 2i
–2 +
2i
–
 
x =
2
2
2
x = –1 +
– i
ANSWER
Substitute values in the
quadratic formula: a = 1,
b = 2, and c = 2.
Simplify.
Simplify and rewrite
using the imaginary unit i.
Simplify.
The solutions are – 1 + i and – 1 – i.
Use the Quadratic Formula
Use the quadratic formula to solve the equation.
2 – 4ac
+
–
b
b
2
–
2x - x = - 4
x =
2a
Rewrite in standard form: 2x2 – x + 4 = 0
−(−1)± (−1)2 −4(2)(4)
x=
2(2)
1± −31
x=
4
1± 31
x=
4
1
4
x= ±
 31
4
Homework
WS 2.6
#1, 2, 4-14even
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