1213141516Factoring Flashcards
Find the x-intercepts of
f(x) = 10x² - 8x
Find greatest common factor:
2x(5x - 4)
intercepts are when either piece = 0
x = 0 or 4/5
What are the x-intercepts, y-intercept and vertex of:
x² - 11x + 18 = 0
x² - 11x + 18 = 0
x-intercepts: (x - 9)(x - 2) → 9,0 and 9,2
vertex: a = 1; b = -11; c = 18 →
x-vertex = 11 /2 :: y = -12.5
y-intercept = 18
What are the x-intercepts, y-intercept and vertex of:
2x² + 8x - 20 = 5x
2x² + 8x - 20 = 5x
2x² + 3x - 20 = 0
(2x - 5)(x + 4) = 0
x-intercepts: (5/2,0) and (-4,0)
y-intercept: (0,-20)
x-vertex: -8/4 = -2
y-vertex: -18
14Factor:
3x² - 48 = 0
Create a difference of 2 squares:
3(x² - 16) = 0
3(x - 4)(x + 4) = 0
Find roots:
4x² + 36x = 88
4x² + 36x - 88 = 0
4(x² + 9x - 22) = 0
4(x + 11)(x - 2) = 0
roots (x-intercepts) = (-11,0) and (2, 0)
What is the square root method for solving quadratic equations?
x² - 36 = 0
x² = 36
x = ±6
What is the square root of 36?
It is only 6, not -6.
The square root is always a positive number.
But, with x² - 36 = 0, either +6 or -6 solves it. In this case you are not looking for a square root.
Simplify √45
√(9*5)
3√5
What is the square root of a negative number?
an imaginary number
a + bi where i = √-1
Solve the equation:
x² - 144 = 0
x² - 144 = 0
x² = 144
√x² = ±√144
x = ±12
We are just solving the equation, not finding the square root of x²
Solve:
x² - 44 = 20
x² - 44 = 20
x² = 64
√x² = ±√64
x = 8 and -8
Solve:
(x - 3)² - 49 = 0
(x - 3)² - 49 = 0
√(x - 3)² = ±√49
x - 3 = ±7
x = 10 or -4
Solve:
x² - 63 = 0
x² - 63 = 0
x² = 63
x = ±√63
x = ±3√7
Solve:
x² + 48 = 0
x² + 48 = 0
x² = -48
x = ±√(16*3)i
x = ±4√3i
What is completing the square?
Manipulating the equation by creating a perfect square.
What will it give you?
The vertex.
Complete the square and find the vertex:
x² + 6x + 42 = 0
x² + 6x + 42 = 0
Divide the 6 by 2 and square it.
(x² + 6x + 9) + 42 - 9 = 0
(x + 3)² + 33 = 0
vertex-x = -6/2 = -3
:: vertex-y = 33 (plug-in)
y-intercept comes from the original equation → 42
What is the vertex of:
y = (x - 5)² + 10
y = (x - 5)² + 10
(5, 10)
When x - 5, (x - 5)² is 0
What are the vertex and intercepts of:
y = x² - 10x - 30
y = x² - 10x - 30
y-intercept = -30
y = (x² - 10x + 25) - 30 - 25
y = (x - 5)² - 55
vertex = (5, -55)
a = 1; b = -10; c = -30
10±√(100 - - 120)/2
5±√220/2
x-intercepts: 5±2√55/2 = 5±√55
Solve:
y = x² + 7x - 8 = 0
y = x² + 7x - 8 = 0
y = (x² + 7x + 7/2²) - 8 -7/2²
(x + 7/2)² - 81 /4
(x + 7/2)² = ± 81 /4
x + 7/2 = ±9/2
x = 1 and -8
Solve:
-x² + 8x = -40
-x² + 8x = -40
faces down
x² - 8x - 40 = 0
y-intercept = 40
(x² - 8x + 16) - 40 - 16 = 0
(x - 4)² - 56 = 0
vertex = (4, 56)
(x - 4) = ±√56
x-intercepts = (4+2√14,0) and (4-2√14, 0)
16Solve:
2x² - 12x - 14 = 0
2x² - 12x - 14 = 0
When you make the equation = 0, you are looking for the x-intercepts (where y = 0)
2(x² - 6x - 7) = 0
Divide both sides by 2:
x² - 6x - 7 = 0
y-intercept = 0
(x² - 6x + 9) - 7 - 9 = 0
(x - 3)² - 16 = 0
vertex = (3, -32) → -16*2
(x - 3) = ±√16
x-intercepts = (-1,0) and (7,0)
What does the Quadratic Formula tell you?
It helps you find the roots of a quadratic equation when you can’t do simple factoring.
What is the Quadratic formula?
For ax² + bx + c:
-b±√(b² - 4ac)/2a
What is b² - 4ac called?
What does it tell you?
The discriminant.
If it is positive, there are 2 x-intercepts.
If it is 0, there are double roots (it sits on the x-axis)
If it is negative, there are no real roots (it does not cross the x-axis)
What does the Quadratic formula tell you about the vertex?
It tells you the x-value of the vertex.
-b±√(b² - 4ac)/2a
-b/2a = x-value vertex
How do you solve a Quadratic Inequality?
Graph the = equation and test points inside and out.
