math

the expression (3/2x+1)(3/2x-1)-(3/2x-1)^2 is equivalent to?

\(\begin{array}{ll} \left( \frac32x+1 \right) \left( \frac32x-1 \right) - \left( \frac32x-1 \right)^2 \\\\ = \left( \frac32x+1 \right) \left( \frac32x-1 \right) - \left( \frac32x-1 \right)\left( \frac32x-1 \right) \qquad & | \qquad \text{factorise } \left( \frac32x-1 \right) \\\\ = \left( \frac32x-1 \right) \left[ \left( \frac32x+1 \right) - \left( \frac32x-1 \right) \right]\\\\ = \left( \frac32x-1 \right) \left( \frac32x+1 - \frac32x+1 \right)\\\\ = \left( \frac32x-1 \right) \left( 2 \right)\\\\ = 2 \left( \frac32x-1 \right) \qquad & | \qquad \text{expand} \\\\ = 2\cdot \frac32x-2\cdot 1 \\\\ = 3x-2 \\\\ \mathbf{ \left( \frac32x+1 \right) \left( \frac32x-1 \right) - \left( \frac32x-1 \right)^2 } \mathbf{ = 3x-2 } \end{array}\)

Simplify the following:
(3/2 x+1) ((3 x)/(2)-1)-((3 x)/(2)-1)^2

Put each term in (3 x)/(2)-1 over the common denominator 2: (3 x)/(2)-1  =  (3 x)/2-2/2:
(3 x)/2-2/2 (3/2 x+1)-((3 x)/(2)-1)^2

(3 x)/2-2/2 = (3 x-2)/2:
(3 x-2)/2 (3/2 x+1)-((3 x)/(2)-1)^2

Put each term in 3/2 x+1 over the common denominator 2: 3/2 x+1  =  (3 x)/2+2/2:
((3 x)/2+2/2 (3 x-2))/(2)-((3 x)/(2)-1)^2

(3 x)/2+2/2 = (3 x+2)/2:
((3 x+2)/2 (3 x-2))/(2)-((3 x)/(2)-1)^2

Put each term in (3 x)/(2)-1 over the common denominator 2: (3 x)/(2)-1  =  (3 x)/2-2/2:
((3 x+2) (3 x-2))/(2×2)-(3 x)/2-2/2^2

(3 x)/2-2/2 = (3 x-2)/2:
((3 x+2) (3 x-2))/(2×2)-(3 x-2)/2^2

Multiply each exponent in (3 x-2)/2 by 2:
((3 x+2) (3 x-2))/(2×2)-((3 x-2)^2)/(2^2)

2^2 = 4:
((3 x+2) (3 x-2))/(2×2)-(3 x-2)^2/4

Combine powers. ((3 x+2) (3 x-2))/(2×2) = 2^(-(1+1)) (3 x-2) (3 x+2):
((3 x-2) (3 x+2))/2^(1+1)-(3 x-2)^2/4

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

((3 x-2) (3 x+2))/2^2-(3 x-2)^2/4 = ((3 x-2) (3 x+2)-(3 x-2)^2)/4:
((3 x-2) (3 x+2)-(3 x-2)^2)/4

(3 x-2) (3 x+2) = (3 x) (3 x) + (3 x) (2) + (-2) (3 x) + (-2) (2) = 9 x^2+6 x-6 x-4 = 9 x^2-4:
(9 x^2-4-(3 x-2)^2)/4

(3 x-2) (3 x-2) = (3 x) (3 x) + (3 x) (-2) + (-2) (3 x) + (-2) (-2) = 9 x^2-6 x-6 x+4 = 9 x^2-12 x+4:
(-9 x^2-12 x+4+9 x^2-4)/4

-(9 x^2-12 x+4) = -9 x^2+12 x-4:
(-9 x^2+12 x-4+9 x^2-4)/4

Grouping like terms, 9 x^2-9 x^2+12 x-4-4 = 12 x+(-4-4)+(9 x^2-9 x^2):
(12 x+(-4-4)+(9 x^2-9 x^2))/4

9 x^2-9 x^2 = 0:
(12 x+(-4-4))/4

-4-4 = -8:
(12 x+-8)/4

Factor 4 out of 12 x-8:
4 (3 x-2)/4

(4 (3 x-2))/4 = 4/4×(3 x-2) = 3 x-2:
Answer: | 3x - 2