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This is the equation of a straight line with slope, m, and y-intercept, b.

108 x 1/3 =108/3=36

Solve for x:
sqrt(x-1) = x

Raise both sides to the power of two:
x-1 = x^2

Subtract x^2 from both sides:
-x^2+x-1 = 0

Multiply both sides by -1:
x^2-x+1 = 0

Subtract 1 from both sides:
x^2-x = -1

Add 1/4 to both sides:
x^2-x+1/4 = -3/4

Write the left hand side as a square:
(x-1/2)^2 = -3/4

Take the square root of both sides:
x-1/2 = (i sqrt(3))/2 or x-1/2 = 1/2 (-i) sqrt(3)

Add 1/2 to both sides:
x = 1/2+(i sqrt(3))/2 or x-1/2 = 1/2 (-i) sqrt(3)

Add 1/2 to both sides:
Answer: |  x = 1/2+(i sqrt(3))/2     or      x = 1/2-(i sqrt(3))/2

2)  g(x) = x^4 - 1

x^4=1

roots    1, i, -1, -i

because 1 is 'primary' one and there will be 4 altogether,  each will be 90 degrees apart on the complex plane so the answer was easy.

4/5=.80 x 100 =80%

1) f(x) = x^3 - 1 

\(x^3-1=0\\x^3=1\\ x=1\\ 3 \;roots\\ 2\pi/3\\ x=e^{(2\pi/3)i}, \qquad x=e^{(4\pi/3)i} \qquad and \qquad x=1 \)

1) 

Solve for x:
x^3-1 = 0

Add 1 to both sides:
x^3 = 1

Taking cube roots gives 1 times the third roots of unity:
Answer: |  x = 1    or     x = -(-1)^(1/3)    or    x = (-1)^(2/3)

2)

Solve for x:
x^4-1 = 0

Add 1 to both sides:
x^4 = 1

Taking 4^th roots gives 1 times the 4^th roots of unity:
Answer: |  x = -1    or    x = -i    or    x = i    or    x = 1

3)

Solve for x:
x^3-2 x^2-5 x+6 = 0

The left hand side factors into a product with three terms:
(x-3) (x-1) (x+2) = 0

Split into three equations:
x-3 = 0 or x-1 = 0 or x+2 = 0

Add 3 to both sides:
x = 3 or x-1 = 0 or x+2 = 0

Add 1 to both sides:
x = 3 or x = 1 or x+2 = 0

Subtract 2 from both sides:
Answer: |  x = 3    or    x = 1    or    x = -2

4)

Solve for x:
x^4+2 x^3-16 x^2-2 x+15 = 0

The left hand side factors into a product with four terms:
(x-3) (x-1) (x+1) (x+5) = 0

Split into four equations:
x-3 = 0 or x-1 = 0 or x+1 = 0 or x+5 = 0

Add 3 to both sides:
x = 3 or x-1 = 0 or x+1 = 0 or x+5 = 0

Add 1 to both sides:
x = 3 or x = 1 or x+1 = 0 or x+5 = 0

Subtract 1 from both sides:
x = 3 or x = 1 or x = -1 or x+5 = 0

Subtract 5 from both sides:
Answer: |  x = 3    or    x = 1    or    x = -1    or    x = -5

This is a geometric sequence because:

3 x 2 = 6

6 x 2 = 12

12 x 2 = 24

24 x 2 = 48

\(-3(x+2)=9(x-6)\)

\(-3x-6=9x-54\)

\(12x=48\)

\(x = 4\)

I will assume that equation is  y = 3/5x-7     ONE perpindicular would be   - 5/3x-7

Oops. I was wrong

\(\sqrt{x-1}=x\)

\(x-1=x^2\)

\(x^2-x+1=0\)

\(x=\frac{-(-1)\pm\sqrt{(-1)^2-4(1)(1)}}{2(1)}\)<------ Quadratic Equation

=\(\frac{1}{2}\pm\frac{\sqrt{-3}}{2}\)

=\(\frac{1\pm\sqrt3i}{2}\)

No real roots and 2 imaginary roots

\(4\times2^{-4}+5\)

=\(\frac{4}{16}+5\)

=\(5\frac{1}{4}\)

or \(\frac{21}{4}\)

Maths is first invented by people of India. They used numbers to count how much of things they have. Soon, there have been problems like 'if you have 3 buckets of oil and people give you 2 buckets of oil how much buckets of oil will you have' Then addition and subtraction is created to calculate. Then, people thought of a good way to calculate problems with addition faster, then there exists multiplication, that's the way you learn maths right like you learn addition, then subtraction, then multiplication ......

Long ago in Greek(Greek? I am not sure) a student of the Great Pythagoras(which is Hippasus)(XDDDDD) found a number that cannot be represented by a simple fraction but at that time people always believed every number can be represented by a fraction which is said by Pythagoras. Soon Pythagoras killed Hippasus because he discovered the number that cannot be represented by a fraction. But the number that was found by Hippasus was quickly known by all people. That's how irrational number existed. Soon, Greek mathematician and engineer wondered what sqrt(-1) is. He can't find any solutions for the equation x = sqrt(-1). So as the number is imaginary, the mathematician set that sqrt(-1) = i. That's how imaginary numbers existed.

Long long time later in Europe, Issac Newton wondered how could he solve a physics question so that he invented calculus.

You should check your results Max!  

You haven't done the quadratic solution correctly.

\(1! = 1\times(1-1)!\)

\(1! = 0!\)

Therefore 0! = 1

\(\frac{2}{5}\times60\)

= \(\frac{120}{5}\)

=24

\(\frac{\frac{35}{36}}{\frac{3}{4}}\)If you can't see zoom in for it

= \(\frac{35}{36}\times\frac{4}{3}\)

=\(\frac{35}{27}\)

50/3 hours

\(sin \vartheta = \frac{5}{12}\)

\(\vartheta = sin^{-1}(\frac{5}{12})\)

which is approximately 24.6243°

Therefore, the missing angle is 24.6243° which is about 0.429775 radians (all corr. to 6 sig. fig.)

\(\sqrt{x-1}=x\)

\(x-1=x^2\)

\(x^2-x+1=0\)

\(x=\frac{(-1)^2\pm\sqrt{4(1)(1)}}{2(1)}\)<------ Quadratic Equation

\(= \frac{1}{2}\pm\frac{2}{2}\)

x = 3/2 or x=-1/2

There are two solutions.

16.6667

1.2962962963

16.6667