Acadec Math

Linear and Quadratic Equations

100

3(2x−1)=4x+5

x=4

200

(x−3)2=7

x=3±√7

300

x²+kx+9=0

k=±6

400

A quadratic equation has no real solutions and a leading coefficient of 1. What must be true about its discriminant, and why?

The discriminant is negative; the parabola does not intersect the x-axis

500

2x²−3x=5x−6

x=2 and x=−2/3

Polynomials and Complex Numbers

100

(3x²+2x)+(x²−5x)

4x²−3x

200

(3x+5)(2x−2)

6x²+4x-10

300

According to the Factor Theorem, what does it mean if f(2)=0f(2) = 0f(2)=0?

(x−2) is a factor of the polynomial

400

Simplify (2+3i)(1−i)

5+i

500

Divide 4+2i/1-i

1+3i

Functions and Inverses

100

What is the definition of a function?

each input has exactly one output

200

Is x=y² a function?

nope

300

Find the inverse of f(x)= 2x+1

f^-1(x)= 1/2x - 1/2

400

How would you determine if a function has an inverse?

If it is one-to-one; each output corresponds to exactly one input

500

If f(g(x))=x, what relationship do f and g have?

They are inverse functions 

Non-polynomial Equations

100

Solve and state restrictions: (x+2)/(x-1) = 3

x=5/2, x≠1

200

Solve √(x+5) = x-1

x=4 (check for extraneous)

300

(4x+2)/5 = ((3-x)/6) + 2

x=3

400

ln(x) = ln(6-x)

x=3

500

Solve logx + log(2x+5) - log7 = 0

x=1

Trigonometry

100

sin(5π/3)

-√3/2

200

Simplify sinx/cscx + cosx/secx

sin²x + cos²x = 1

300

Solve on [0, 2π): sin(2x) = √3/2

x = π/6,  π/3,  7π/6,  4π/3

400

Simplify sinx(1+cosx) / 1-cos²x

cscx + cotx

500

If secθ=-5 and sinθ>0, find cscθ

(5√6)/12