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1) 15x^2-11x-14=0

Multiply 15 and -14 to get -210.

Now look at factors or -210 that add up to -11.

The factors are -21 and +10.

Why? Because -21*10 = -210 and -21+10 = -11

Replace -11x with -21x+10x

15x^2 – 21x + 10x – 14 = 0

Factor by grouping the first two terms and grouping the last two terms.

3x(5x-7) + 2(5x-7) = 0

(5x-7)(3x+2) =0

5x-7 = 0 or 3x+2 = 0

x = 7/5 or x = -2/3

2) 4(3x-1)^2=9

Take the square root of each side (remember that when you take the square root of 9, you will get two values (+ or -)).

2(3x-1) = 3 or 2(3x-1) = -3

Solve each equation separately to find the two solutions.

3) 2x^2-8x+1=0

Subtract 1 from each side

2x^2 – 8x = -1

Divide both sides by 2

x^2 – 8x = -1/2

Take the coefficient of x, divide it by 2, then square it. Add this number to both sides.

x^2 – 8x + 16 = -1/2 + 16

(x-4)^2 = 31/2

Now solve by the method of #2.

4) 2x^2-3x-20=0

The quadratic formula says that the solutions of the quadratic equation ax^2 + bx + c = 0 are

x = [-b +/- sqrt(b^2 -4ac)] / (2a)

So a = 2, b = -3, and c = -20.

Plug in those values and simplify.

Last one.

5) 2i^74+5i^105

Let’s look at powers of i.

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

i^5 = i

i^6 = -1

i^7 = -i

i^8 = 1

Every four cycles i repeats itself. Also, notice that the power of i is never greater than one. If it is, then that means you can simplify it.

To simplify, divide the exponent by 4 and look at the remainder.

74/4 = 18 remainder 2

That means i^74 = i^2 = -1

105/4 = 26 remainder 1

That means i^105 = i^1 = i

So 2i^74+5i^105

= 2(-1) + 5(i)

= -2 + 5i