Show algebraically that for positive integers pp and qq, if
a = p2 − q2
b = 2p
c = p2 + q2
then a2 + b2 = c2.

a² + b² = c²

⇒ p⁴ + q⁴ + 2p²q² = p⁴ + q⁴ + 2p²q²

Step-by-step explanation:

Data provided:

a = p² − q²

b = 2p q                   (for the required result the value of b should be '2pq' instead of '2p' )

c = p² + q²

To show:

a² + b² = c²

Now,

a² = (p² − q² )²

we know,

(x + y) = x² + y² + 2xy

(x - y) = x² + y² - 2xy

Thus,

a² = (p² − q² )²

=  (p²)² + (q²)² - 2p²y²

= p⁴ + q⁴ - 2p²q²       (1)

and,

b² = (2pq)² = 4p²q²         (2)

also,

c² =   (p² + q² )²

= (p²)² + (q²)² + 2p²y²

= p⁴ + q⁴ + 2p²q² (3)

Now,

adding 1 and 2, we get

a² + b² =  p⁴ + q⁴ - 2p²q² + 4p²q²

or

a² + b² =  p⁴ + q⁴ + 2p²q²         (4)

equation (3) and (4), we get

a² + b² = c²

⇒ p⁴ + q⁴ + 2p²q² = p⁴ + q⁴ + 2p²q²

d

step-by-step explanation:

the multiples of 15 are 15, 30,45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195,   210, 225, 240, 255, 270, 285, 300, 315, 330, 345, 360, 375,   390, 405, 420, 435, 450, 465, 480, 495, 510, 525, 540, 555, 570, 585, 600, 615, 630, 645, 660, 675, 690, 705, 720, and 735.

the factors of 15 are 1, 3, 5, and 15.

hope that , fam!

-une, your fren : )