Hi,

Let me give more detailed explanation for the first equation and then I will solve the others with less detail.

a.

x^2 - 2x - 3 = 0

Take the constant term to the right side of the equation,

x^2 - 2x = 3

Multiply each term in the equation by four times the coefficient of the x^2 term,

4x^2 - 8x = 12

square the coefficient of the original x term and add it to both sides of the equation,

4x^2 - 8x + 4 = 12 + 4

4x^2 - 8x + 4 = 16

left side of the equation is a perfect square,

(2x - 2)^2 = 16

take the squareroot of both sides,

Hello,

2x - 2 = -4 or 2x - 2 = 4

2x = -2 or 2x = 6

x = -1 or x = 3

So the solution for the first part of your question is,

x = -1, 3

Part b.

4x^2 - 4x - 3 = 0

4x^2 - 4x = 3

this is great, I was getting hung up on the 4 step subtracting the 4 instead of adding.

64x^2 - 64x = 48

64x^2 - 64x + 16 = 48 + 16

(8x - 4)^2 = 64

8x - 4 = -8 or 8x - 4 = 8

8x = -4 or 8x = 12

x = -4/8 or x = 12/8

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

Part c.

x^2 + 12x - 64 = 0

x^2 + 12x = 64

4x^2 + 48x = 256

4x^2 + 48x + 144 = 256 + 144

(2x + 12)^2 = 400

2x + 12 = -20 or 2x + 12 = 20

2x = -32 or 2x = 8

x = -16 or x = 4

Part d.

2x^2 - 3x - 5 = 0

2x^2 - 3x = 5

16x^2 - 24x = 40

16x^2 - 24x + 9 = 40 + 9

(4x - 3)^2 = 49

4x - 3 = -7 or 4x - 3 = 7

4x = -4 or 4x = 10

x = -4/4 or x = 10/4

x = -1 or x = 5/2

Let me know if you have any questions.

Thank you for going through this, the errors I had were simple, but make a big difference. I'm learning through the comparison.

That's very good.