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Name
Period
ALGEBRA 2
Unit 2 –
Solving Quadratics
Completing the Square
Essential Question: “
“
Perfect Square Trinomials
Factor each expression.
𝑥 2 + 10𝑥 + 25 =
𝑥 2 + 18𝑥 + 81 =
Observation
𝑏 2
𝑎𝑥 + 2(√𝑐)𝑥 + 𝑐 = 𝑎𝑥 + 𝑏𝑥 + ( )
2
2
2
Middle coefficient =
End term =
What number would complete the square giving us a perfect trinomial?
𝑥 2 + 14𝑥 + _____ =
𝑥 2 + 24𝑥 + __________ =
3) 𝑥 2 + 3𝑥 + ___________ = 0
Completing the Square

Given the standard form
vertex form of the equation
, we complete the square to find the
Completing the Square
1) Move _________________________ to the other side of the equation
2) ______________________ to both sides of the equation
3) Find ____________ and place in each blank
4) ________________ left side of the equation and _______________ right side of the
equation
 Factors 
5) ________________ factors. (one factor squared)
6) Take the ______________________ of both sides
7) ______________ for variable
1. Solve by completing the square.
𝑥 2 − 2𝑥 − 5 = 0
Move c to the opposite side and add blanks.
Fill in the blanks by completing the square.
Factor the trinomial.
Condense the factors.
Solve for x.
2. Solve by completing the square.
𝑥 2 − 12𝑥 = −5
Completing the Square when 𝑎 ≠ 1

When there is a leading coefficient, still move c to the opposite side, but then we must
factor the coefficient out. This adds an extra step!
Solve by completing the square.
4𝑥 2 = 3 − 4𝑥
Complex Numbers Discovery
Use your book (pg. 274-275) to define the following words and to answer any of the questions.
Imaginary Number -
i=
𝑖2 =
How can I rewrite the square root of a negative number?
Example Rewrite √−4 using imaginary numbers.
Complex Number -
What is the standard form for a complex number?
Absolute Value of a Complex Number -
|𝑎 + 𝑏𝑖| =
Complex Numbers
Essential Question: “
“
Imaginary Numbers
𝒊=
𝒊𝟐 =
𝒊𝟑 =
𝒊𝟒 =
* Imaginary numbers are
*
When simplifying imaginary numbers:


𝑖5 =
𝑖7 =
𝑖 12 =
To convert imaginary numbers:
 Divide exponent by four
 Remainder will tell us the power of i
o Remainder of 1 =
o Remainder of 2 =
o Remainder of 3 =
o Remainder of 4 =
Examples:
1. 𝑖 63 =
2. 𝑖 120 =
3. 𝑖 7 =
4. 𝑖 36 =
5. 𝑖 15 =
6. 𝑖 86 =
Simplify the following expressions.
1. 𝑖 4 + 𝑖 8 + 1 =
2. 2𝑖 − 6𝑖 2 + 5𝑖 14 + 7𝑖 9 =
3. 10𝑖 2 + 4𝑖 4 + 6 =
4. 12𝑖 − 10𝑖 22 + 8𝑖 16 − 6𝑖 33 + 16𝑖 80 =
Complex Numbers
Standard Form:
Examples of Complex Numbers:
Square Root of a Negative Number
For any number, n,
√−𝒏 = 𝒊√𝒏
Rewrite the following negative roots as roots with imaginary numbers.
√−4 =
√−13 =
√−25 =
Complex Number Operations
Essential Question: “
“
Solve each of the equations.
1) (5𝑥 + 4)(3𝑥 + 12) =
2) (7 − 13𝑥) + (9 + 16𝑥) =
3) (4 + 7𝑥) − (12 − 16𝑥) =
4) (4𝑥)(7𝑥) =
5) (6𝑥 + 4)(3𝑥 + 10) =
Adding Complex Numbers
1) (5 + 7𝑖) + (−2 + 6𝑖) =
2) (10 − 2𝑖 2 ) + (−8 + 4𝑖 2 ) =
3) (5 + √−13) + (8 + √−12) =
Subtracting Complex Numbers
1) 7 – (3 + 2i)
2) (4 – 6i) – (2 + 3i)
3) (12 + 6i) – 10i
Multiplying Complex Numbers
1) (5i)(-4i)
2) -6(2 – 4i)
3) 2i(3 – 6i)
4) (2 + 3i)(3 + 5i)
5) (5 – 2i)(-4 + i)
Finding Complex Solutions



Get the
Take the
Simplify the
1) 4𝑥 2 + 100 = 0
by itself
of each side
2) 𝑥 2 = −16
Quadratic Formula
Essential Question: “
“
The solution for a quadratic equation is values of x that
SOLUTION =
.
=
Quadratic Formula

The equation must be in the standard form
Standard form:
X=
a–
b–
c–
Solve 3𝑥 2 + 5𝑥 + 1 = 0 using the quadratic formula.
a=
b=
c=
Substitute a, b, and c into the quadratic formula.
𝑥=
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
=
Solve 𝑥 2 − 𝑥 − 1 = 0 using the quadratic formula.
Solve 3𝑥 2 + 2𝑥 + 4 = 0 using the quadratic formula.
Discriminant
Discriminant =
Value of the Discriminant
Type and number of solutions
for
𝒂𝒙𝟐 + 𝒃𝒙 + 𝒄 = 𝟎
𝑏 2 − 4𝑎𝑐 > 0
𝑏 2 − 4𝑎𝑐 = 0
𝑏 2 − 4𝑎𝑐 < 0
Determine the type and number of solutions for 𝑥 2 − 6𝑥 + 7 = 0.
a=
b=
c=
Substitute a, b, and c into discriminant.
Determine the type and number of solutions for 4𝑥 2 + 2𝑥 + 3 = 0.