Graphing
Transformations of the equation y=(x-1)2+2
What are up 2 and right 1?
Solving x2 -64=0
(x-8)(x+8)
What is x=-8,8 ?
4x2=81
x2=20.25
What is x= +/- 4.5
Value of c in Solve x2+10x +c
25
Use the discriminate to find the number of roots of 2x2 -3x +1 = 0
2x2-3x+1 =0
(-3)2 -4*2*1 = 9-8 = 1
What is 2 real roots?
Domain of the graph y= 2x2 +1
What is all real numbers? or (-infinity, infinity)
Solve x2-10x+24 =0
(x-6)(x-4)
What is x=6,4 ?
5-6y2=113
-6y2=108
y2=-18
y= +/- sqrt(-18)
y= +/- 3i sqrt 2
Solve x2+12x +32=0
x2+12x = - 32
x2+12x +36 = -32 +36
(x+6)2 = 4
x+6 = +/- 2
x= -6 +/- 2
What is x= -8, -4
Discriminate of x2+4x =-7 to determine number of roots
x2+4x+7 =0
42 -4*1*7 = 16-28 = -12
What is 2 imaginary roots?
The range of the function y= -3(x-2)2 + 5
What is (y< 5) or (-infinity, to 5]
Solve 2x2+3x+1=0
(2x +1)(x+1)
What is x= -1/2, -1
(x-5)2 =36
x-5 = +/- 6
x= 5 +/- 6
x= 11, -1
Solve x2+10x =-8
x2+10x +25 =-8 +25
(x+5)2= 17
x+5 = +/- sqrt(17)
x= -5 +/- sqrt(17)
Solve 2x2 +5x =-3
2x2 +5x +3 =0
x = -1, -1.5
vertex of y= x2+4x+3
2 Methods:
1: Factor (x+3)(x+1): Roots x=-1, -3 so halfway between them x=-2 is the axis of symmetry plug in x=-2 y=(-2+3)(-2+1) =-1 so (-2,-1)
2: Use axis of symmetry formula x=-b/2a = -4/2*1 =-2 plug back in y= (-2)2 +4(-2)+3 = -1 so (-2,-1)
What is (-2,-1)?
Solve 3x2+11x +10
(3x + 5)(x + 2)
WHat is x=-5/2, -2
(4x-3)2+ 7 =39
(4x-3)2=32
4x-3 = +/- sqrt(32)
4x-3 = +/- 4sqrt(2)
x= 3 +/- 4sqrt(2) divided by 4
2x2-4x =6
2(x2-2x + c) = 6+ 2c
(b/2)2 = 1
2(x2-2x + 1) = 6+ 2(1)
2( x-1)2=8
( x-1)2=4
x-1= +/- 2
x= 1 +/- 2
x= 3, -1
Solve 2x2-6 =-x
2x2+x-6 =0
x= -2, 1.5
Change the equation into vertex form y=x2 +4x+16
Complete the square y= (x2+4x +c)+16 -c
c= (b/2)2 = (4/2)2=4
y= (x2+4x +4)+16 -4
What is y= (x+2)2+12 ?
Factor 3x3+15x2+18x
3x( x2+x-6)
3x(x-2)(x+3)
x=0, 2,-3
Why you cant solve this problem by square roots
x2+7x=0
What is the term bx?
Explain the process of completing the square
Set the equation up for example 2x2 +4x =8
Divide by a so the 2(x2 + 2x ) = 8
Complete the square by finding (b/2)2
2(x2 + 2x + 1) = 8+ 2
Write left side as its factored form 2(x+1)2=10
Solve the rest of the way
It is said all problems can be solved by quadratic formula.
Solve: (x-5)2=20
* Hint: Expand the equation
Expand the equation: (x-5)(x-5)=20
x^2 -10x+25 =20
x^2-10x-5 =0
Plug in to the formula:
x=.528, 9.47