x^2 + x = 3/4?

x^2 + x = 3/4
Complete the square
(1/2b)^2, then add to both sides
b = coefficient of x, which is 1
1/2^2
1/4
x^2 + x + 1/4 = 3/4 + 1/4
(x + 1/2)(x + 1/2) = 1
Take SquareRoot of both sides
x + 1/2 = 1
x = 1/2

4x² + 4x - 3 = 0
x = [ - 4 ± √ (16 + 48 ) ] / 8
x = [ - 4 ± √ (64) ] / 8
x = [ - 4 ± 8 ] / 8
x = 1/2 , x = - 3/2

x^2 + x + 1/4 - 1/4 = 3/4
(x+1/2)^2 = 3/4 + 1/4
(x+1/2)^2 -1 =0
(x + 1/2 - 1) * (x+1/2 +1) = 0
(x -1/2) * ( x + 3/2) = 0
x = 1/2 or x = -3/2

one can use completing the square or in fact use the quadratic formula which is a formula based on completing the square for the general case aX^2 + bX + C = 0

the quadratic formula is X = -b +/- SQRT ( b^2-4ac) all over 2 a

so -1 +/- SQRT( 1 -4(1)(-3/4)) -1 +/- SQRT( 4)/2
X = 1/2 , -3/2

x^2 + x = 3/4
4(x^2 + x) = 3
4x^2 + 4x - 3 = 0
(2x - 1)(2x + 3) = 0
x = 1/2 or x = -3/2

x^2 + x = 3/4
x^2 + x - 3/4 = 0
x^2 + 3x/2 - x/2 - 3/4 = 0
(x)(x + 3/2) - (1/2)(x + 3/2) = 0
(x + 3/2)(x - 1/2) = 0

x + 3/2 = 0
x = -3/2 (-1.5)

x - 1/2 = 0
x = 1/2 (0.5)

∴ x = -3/2 (-1.5) , 1/2 (0.5)

You can format this so you put it into the quadratic formula, or complete the square, or multiply all by 4 to see if things factor. I'll do the complete the square since it's already set up for it:

x² + x = 3/4
x² + x + 1/4 = 3/4 + 1/4
(x + 1/2)² = 1
x + 1/2 = ±√1
x + 1/2 = ±1
x = -1/2 ± 1
x = -3/2, 1/2