Math Final

Vertex is the x coordinate

- b/2a

what is quadratic formula?

-b (+ and -) √b - 4ac / 2a

Simpligy: x-4/x^2 - 25 / 1 + 1/x-5

1 / x+5

Completing the square of: x^2 - 8x + 2= 0

1) divide everything by a (if possible) 2) move all constant terms to the right 3) divide x term B by 2 then square it and add your answer to both sides of the equation 4) factor left 4) apply square root property answer: {4 - √14, 4 √14}

Determine number and nature of the solutions: 1) 1/2 x^2 + 5x + 9/2 2) 2x^2 + 8x + 8

1) two real solutions 2) exactly one real solution

Nina owns a fragrant shop and needs to make 6 ounces of 30% sandalwood essential oil mixture. She has 10% sandalwood mixture and 40% sandalwood mixture available. How much of each mixture should she combine to create customer's desired product?

2 ounces of 10% and 4 ounces of 40% Why? 10 (x) + 40 (6-x) = 30 (6)

Find product of (2x-3) (x^2 + 4x - 5)

2x^3 + 5x^2 -22x +15

√-1 + 3 - √-25

3 - 4i

For f(x)= 2 |x-8|/ √4x-12 evaluate f(4)

4

Simplify: (- 2 a^4 b^-7 / c^4)^2 (7 a^-3 b^4 / c^-5)

4a^14 / 49 b^22 c^18

24x^3+20x^2-84x

4x(3x+7)(2x-3)

√125x^3y X √2xy^6

5x^2 |y^3| √10y

Subtract and simplify: 14 ∛2 - 3 ∛16

8 ∛2

Multiply the expressions and simplify: x^2 + 13x +42 / 3x + 18 X 24x / x^2 + 6x - 7

8x / x - 1

√81x^2

9 |x|

Which line is horizontal? a) y= 2 b) x=2

A because y = some # and slope= 0

Solve for x: |4x+5| > -6

ALWAYS TRUE so (- infinity, infinity)

Which polynomial has greatest degree? a) 36pq^4 - 180q^6 b) 17 - 10x + 1000 x^7

B

Peter went out to lunch and ate a burger and a 2 shakes for a total of 600 calories. If a burger has 2 times as many calories as one shake, how many calories are in each?

B + 2S = 600 B= 2S answer: a burger has 300 cal and a shake has 150 cal

Which line is vertical? a) y=2 b) x=2

B because x= some # and has undefined slope

Which relation is a function? a) (1,1) (1,2) (1,3 (1,4) b) (0,1) (1,2) (2,1) (3,2)

B) because x values cannot repeat

What is the center and radius of the circle: (x+1)^2 + (y-4)^2 = 25

Center = (-1, 4) Radius= 5

solve x^2 = 2x + 4

Get equation = to 0 then use quadratic formula answer = 1(+ and -) √5

given quadratic equation 2x^2 - 3√x + 4, find the discriminant and determine the number of real solutions

If D > 0 then there are 2 real solutions If D = 0 then there is exactly 1 real solution If D < 0 then there are 2 non-real solutions Answer -29 < 0 so D= -29, 2 non real solutions

are the two lines parallel, perpendicular, coinciding or intersecting? 4x-3y-7 = 0 3x -4y +6 = 0

Intersecting

FInd the midpoint of the line segment from (-3,2) and (4,6)

Midpoint formula= (x1 + y1)/ 2, (x2+y2)/ 2 Answer: (1/2, 4)

Solve for x: |6x-2|< -14

NEVER TRUE so NO SOLUTION

1 / x + 4 + 7 / x + 8 = 4 / x^2 + 12x + 32

NO SOLUTION

Line perpendicular to 2x - y= 4 that passes through point (-1, 1)?

Perpendicular lines have slope = negative reciprocal answer: y= -1/2 x + 1/2

which is true about the linear system: -x + 3y= -7 5x -15y = 4 a) this is inconsistent. no solutions b) this is dependent. infinitely many solutions c) this is independent. there is one solution

a) This is inconsistent. No solutions

Which is an identity? a) (5-x) +5=x b) 3 (m+1)-m= (m+5) - (2-m)

b) 2m +3 = 2m+ 3 (mirror)

Which is a conditional equation? a) 2 (x-3) +1 = 2x-4 b) 2 (1-x)+2= 2x

b) x= 10

are the two lines parallel, perpendicular, coinciding or intersecting? 2x + 3y = 3 y= -2/3 x + 1

coinciding

Write the standard form of the circle: x^2 + y^2 + 2x - 4y +1= 0

complete the square answer: (x+1)^2 + (y-2)^2 = 4

Distance formula?

d (a,b) = √(x2 - x1)^2 + (y2 - y1)^2

Find the x intercepts of the line: 4x - 2y = 8

set x to 0 to find y int and set y to 0 to find x int x int= (2, 0) y int= (0, -4)

Divide polynomial: 3x^4 -4x^2 +1 / x + 1

since you have the form x - c set bottom equal to zero and use synthetic division or divide using long division answer: 3x^3 - 3x^2 -x +1

Equation of horizontal line passing through point (2, -8)?

y= -8

What is the range if the relation: (2,1) (2,3) (2,4) (3,5) (4,7)

{1, 3, 4, 5, 7} range is y values

What is the domain of the relation: (2,1) (2,3) (2,4) (3,5) (4,7)

{2, 3, 4} domain is x values

two equations are the same line. there are infinitley many solutions.

dependent system (consistent)

Write equation: 3y+x^2 = 9 as a function of x

f(x)= 3- x^2/3

Standard form of a quadratic function?

f(x)= a(x-h)^2 + k vertex= (h,k)

Write the equation of a line passing through: (1, 2) and (-2, 5)

find slope first then use point slope formula y-y1= m (x-x1) answer: y= -x+3

Eq of line parallel to 3x-5y= 1 that passes through (1, -2)

get in y= form first parallel lines have SAME SLOPE different intercepts answer: y= 3/5 x - 13/5

i^83 = ?

i^3 = -i

2 equations will meet at one point. when solving you will get one answer

independent system (consistent)

two lines have no common points (they are parallel). when solving you will get no solution.

independent system (inconsistent)

if |a| is > 1 the parabola will be ____ than y=x^2 if |a| is < 1 the parabola will be _____ than y= x^2

narrower wider

Which statement is true about the parabola f(x) = 1/2 (x+2)^2 - 1 A) parabola is narrower than y=x^2 B) parabola is obtained by shifting the graph of y=x^2 right 2 units and down 1 unit C) the y intercept of the paraobola is (0,1) D) the axis of symmetry is x=2

not A bc |a|< 1 it will be wider (if |a|> 1 it is narrower) not b because you take the opposite sign of x+2, which is -2 so you shift left 2 units YES c because if you set equation y= 1/2 (x+2)^2 -1 to zero you get y=1 not d because if you set x to 0 you get -2

Write standard form of circle with radius √5 and center (-2,3)

r^2 = (x-h)^2 + (y-k)^2 **remember (h,k) answer: 5= (x+2)^2 + (y-3)^2

x^2 + c shift ____ c units x^2 - c shift ____ c units (x+c)^2 shift ____c units (x-c)^2 shift____ c units

UP DOWN LEFT RIGHT

Parabola opens _________if x^2 is positive Parabola open ________ if x^2 is negative

Up Down

Vertex of the graph: f(x)= -6x^2 + 24x + 6

Use vertex formula -b/2a then plug that answer into f(x) answer= (2,30)

3 (1/x-1)^2 - 5/x-1 - 2

u: 1/x-1 answer: x= 3/2, x= -2

solve for x: x^2/3 + x^1/3 - 6 = 0

use substitution u= (x^1/3) ^2 u ^2 = X^2/3 answer: x= -27 and x= 8

solve the linear system: 2x - y = -5 3x + 4y = 9

use substitution or elimination x= 2, y= 7

In order to be a function, a graph must pass the ______

vertical line test

Simplify the rational expression: x^3 + 2w^2 - 15w / 2w^4 - 50w ^2

w-3 / 2w (w-5)

Solve the inequality: x^3 - 3x^2 ≤ 10x

(- infinity, -2] U [0,5]

Solve rational inequalityL x^2 - 8 / x + 4 ≤ x

(- infinity, -4) U [-2, infinity)

Find the solution set of the absolute value inequality -|3x+2| +5 > -6

(-13/3, 9)

Solve for x: |2x-7| + 6 > 11

(-infinity, 1) U (6, infinity)

factor 100m^2 - 81n^2

(10m -9n) (10m +9n)

Find the solution set for -9 <8-7x/3 < -2

(2,5]

factor 2b^3 + ab^2 -8b - 4a

(2b +a) (b+2) (b-2)

factor (5y-x) (2y-y) + (3x+1) (2x - y)

(2x - y) (2x +5y +1)

factor 2x^2 + 9x -18

(2x -3) (x+6)

factor 27x^3 - y^3

(3x - y) (9x^2 +3 xy + y^2)

Write standard form of the circle: x^2 + y^2 + 10x + 9=0 what is the center? what is the radius?

(x+5)^2 + y^2 = 16 Center= (h,k)= (-5, 0) Radius=√16 = 4

factor x^6 - y^6

(x^2)^3 - (y^2)^3 (x^2 - y^2) (x^4 + x^2 y^2 + y^4) answer: (x+y) (x-y) (x^4 + x^2 y^2 + y^4)

5- 6i / 4+7i

-22/65 - 13/65 i

given (f)x = 8x^3 - 5x + 6 = and g(x)= -2x^3 + 6x^2 -6x + 13, evaluate (f - g) (-1)

-24

Use quotient rule to simplify: ∛-72m / ∛3m^-2

-2m∛3

Multiply and divide then simplify: 3x^2 + 14x - 5 / x^2 - x - 20 ÷ 3x - 9x^2 / x^2 + 5x + 4 X 15x / x^2 - 25

-5 (x+1) / (x-5)^2

i^19 + i^100

-i +1

When making a long distance call, Steve pays 2.70 for the first 3 minutes. After that each additional minute costs 0.30. How many minutes can Steve call long distance for 6.00?

14 minutes

Working together Kacey and Maria can solve a rubik's cube in 10 minutes. Working alone it takes Maria twice as long as Kacey to solve the rubik's cube. How long will it take Kacey to solve the cube working by herself?

15 minutes

A boat can travel 12 miles upriver in the same amount of time it takes to travel 18 miles downriver. If the speed of the current is 3 mph, how fast can the boat travel in still water?

15 mph

write f(x)= 2x^2 - 4x + 8 in standard form

2 (x-1)^2 + 6 x(h)= 1 and y(k)= 6 and a= 2

What is the discriminant formula?

D= b^2 - 4ac

4^√- 1000 / x^8

NOT REAL

Find the domain of the radical function: f(x)= √x+14

[-14, infinity)

solve the following equation for x: x^2 - 6x + 10= 0

x = 3 + i, x= 3 - i

solve for x: 2 (x+3)^2 + 32 = 0

x= -3 (+ or -) 4i

Solve for x: 2/3 x - 5/4 = 1/6 +5/12 x

x= 17/3

Solve for x: x(x^2 +2) + 8 = 6 (x^2 + x) -4x^2

x= 2, x= -2

Solve for √x + 1 + 5 = x

x= 8

The axis of symmetry occurs at __________ where h is the x coordinate of the vertex

x=h

Standard form of a circle with radius 3 and center (0,0)

x^2 + y^2 = 9

what is equal to (x-2)^3 ?

x^3 - 6x^2 + 12x - 8