SECTION 9.1
The Square Root Property and
Completing the Square
Think back . . .
• Up to this point, we have factored and set the
factors equal to zero.
Example 1
Solve using the zero-factor property.
2
2𝑥 = 9𝑥 − 4
Another Quadratic Technique
𝑥 2 − 49 = 0
Example 2
Solve using the square root property.
2
3𝑥 − 72 = 0
Example 3
Solve using the square root property.
(3𝑚 − 6)2 = 27
Completing the Square
• Completely relies on the concept of a perfect
square trinomial.
(𝑥 + 𝑦)2 = 𝑥 2 + 2𝑥𝑦 + 𝑦 2
• Think back . . .
(𝑥 + 3)
2
What number makes it a perfect square?
2
𝑥 + 6𝑥 + ______
𝑝2 − 12𝑝 + ______
𝑞 2 + 9𝑞 + ______
1
𝑥 2 − 𝑥 + ______
2
Example 4
Solve the equation by completing the square.
2
𝑥 + 8𝑥 + 11 = 0
Example 5
Solve the equation by completing the square.
2
3𝑦 + 2𝑦 + 2 = 4
Example 6
Find the nonreal complex solutions.
2
(𝑥 − 9) = −7
Example 7
Solve.
2
𝑥 + 6𝑥 + 25 = 0
SECTION 9.2
The Quadratic Formula
Quadratic Solving Techniques We’ve Learned
Techniques
Factoring
Square Root Property
Completing the square
Examples
2
𝑥 + 5𝑥 + 6 = 0
𝑥−5 2 =7
2
𝑥 + 4𝑥 − 2 = 0
A Technique for ANY Quadratic
• The Quadratic Formula can be used to solve ANY
quadratic equation.
• You can derive it by completing the square.
• NOTE: The above formula is used when the
equation has been put into standard form.
Example 1
Solve using the quadratic formula.
2
−2𝑥 + 2𝑥 = 1
Example 2
Solve using the quadratic formula.
𝑦−3 𝑦+5 =2
Example 3
Solve using the quadratic formula.
𝑧 2𝑧 + 3 = −2
The Discriminant
• We can use the discriminant to determine the number
and type of solutions.
Example 4
• Use the discriminant to describe the number and type of
solution(s).
a.
2
9𝑥 − 12𝑥 − 1 = 0
b. 3𝑧 2 = 5𝑧 + 2