Solve 2X^2+5X=3 by factorising?

2x^2 + 5x = 3
2x^2 + 5x - 3 = 0
2x^2 + 6x - x - 3 = 0
(2x^2 + 6x) - (x + 3) = 0
2x(x + 3) - 1(x + 3) = 0
(x + 3)(2x - 1) = 0

x + 3 = 0
x = -3

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

∴ x = -3 , 1/2 (0.5)

Initializing:
2x² + 5x = 3
x² + 5/2x = 3/2
x² + 5/4x = 3/2 + (5/4)²
x² + 5/4x = 24/16 + 25/16
(x + 5/4)² = 49/16
x + 5/4 = 7/4

1st factor:
= x + 5/4 - 7/4
= x - 1/2
= 2x - 1

2nd factor:
= x + 5/4 + 7/4
= x + 3

Values of x:
2x - 1 = 0, 2x = 1, x = 1/2
x + 3 = 0, x = - 3

Answer: x = 1/2, - 3; Factors: (2x - 1)(x + 3)

Proof (x = 1/2):
2(1/2)² + 5(1/2) = 3
2(1/4) + 5/2 = 3
1/2 + 5/2 = 3
6/2 = 3
3 = 3

Proof (x = - 3):
2(- 3²) + 5(- 3) = 3
2(9) - 15 = 3
18 - 15 = 3
3 = 3

Proof (going back to original equation: F.O.I.L.):
(2x - 1)(x + 3) = 0
2x² + 6x - x - 3 = 0
2x² + 5x - 3 = 0
2x² + 5x = 3

2x² + 5x - 3 = 0
(2x-----)(x-------) = 0
(2x-----1)(x------3) = 0
(2x - 1)(x + 3) = 0
x = 1/2 , x = - 3

2x^2 + 5x - 3 = 0
(2x - 1)(x + 3) = 0
x=0.5 or -3