Unit 3: Expressions and Equations
Lesson 1: Solving Quadratics Graphically/Solving Quadratics by Factoring (P3)
Special Factoring Patterns
Pattern Name
Pattern
Example
Difference of Two Squares
5) 64​x2 ​- 81
7) 49​n2 ​- 64
​
​
6) 9​b2 ​- 49
8) 16​x2 ​- 9
9) 7​x2 ​– 343
10) 125​x3 ​- 245​x
​
11) 10​x2 ​- 90​
12) 324​x3 ​- 4​x
Special Factoring Patterns
Pattern Name
Pattern
Example
Perfect Square Trinomial
​
​
​
13)
​ 9​x2 ​- 30​x + 25
14) 4​x2 ​+ 4​x + 1
15) 100​x2 ​+ 160​x + 64
16) 20​x3 ​- 60​x2 ​+ 45​x
​
17) ​16​x2 ​- 40​x + 25
18) ​16​v2 ​+ 24​v + 9
19) ​9​a2 ​- 24​a + 16
20) ​25​k2 ​- 10​k + 1
21) ​4​b3 ​+ 24​b2 ​+ 36​b
22) ​125​n3 ​+ 100​n2 ​+ 20​n
23) ​32​n2 ​+ 48​n + 18
24) ​16​x2 ​- 48​x + 36
Solving Quadratics Graphically
1. Solving quadratic equations​ like 3x2 − 15 = 0 and 3x2 − 15x = 0 ​locates the
x-intercepts on the graphs​ of quadratic functions y = 3x2 − 15 and y = 3x2 − 15x .
a. Using the graphs above, explain how the symmetry of these parabolas can be used to relate
the location of the minimum (or maximum) point on the graph of a quadratic function to
the x-intercepts.
b. Write your solutions from Problem 2 ​on the lines provided. Then find the
coordinates of the vertex point on the graphs of these quadratic functions.
i. y = x2 + 4x
x = ____ , _____
ii. y = 3x2 + 10x
x = ____ , _____
iii. y = x2 − 4x
v. y =− 2x2 − 6x
vii. y = x2 − 2x − 3
x = ____ , _____
x = ____ , _____
x=_____,_____
iv. y =− x2 − 5x
x = ____ , _____
vi. y = x2 − x
x = ____ , _____
viii. y = x2 + 5x + 4
x = ____ , _____
Check for Understanding…
Solve the following graphically.
1)
4x2 − 12x = 0
2) x2 + 2x = 0
3) 5x2 − 20x = 0
4) x2 + 5x + 6 = 0
5) x2 + 1x − 6 = 0
6) x2 + 4x + 3 = 0
Solving Quadratics by Factoring
You can use factoring to solve certain ​quadratic equations. A ​quadratic equation ​in one
variable can be written in the form of
form​ of the equation. If the left side of
be solved using the ​zero product property.
Zero Product Property
Solving Quadratic Equations
25)
26)
27)
​ where
.​ This is called ​standard
​ can be factored, then the equation can
28)
29) ​n2 ​+ ​n = 42
30) ​p2 ​+ 28 = -11 ​p
31) 35​a2 ​+ 48 = -86​a
32) 14​x2 ​- 27​x = -9
33) -​b2 ​+ 40​b + 31 = -3​b - 6​b2 ​+ 7
34) 7 ​p2 ​- 17 ​p + 2 = -2 ​p