√m + √n = 9 ← this is the sum S √m * √n = 9 ← this is the product P √m & √n are the solution of the following equation: x² - Sx + P = 0 → where S is the sum and where P is the product x² - 9x + 9 = 0 Δ = (- 9)² - (4 * 9) = 81 - 36 = 45 x = (9 ± √45)/2 x = (9 ± 3√5)/2 ← these are √m & √n First case: √m = (9 + 3√5)/2 (√m)² = [(9 + 3√5)/2]² m = (9 + 3√5)²/4 m = (81 + 54√5 + 45)/4 m = (126 + 54√5)/4 m = (63 + 27√5)/2 m = 9.(7 + 3√5)/2 → then you can deduce that: n = 9.(7 - 3√5)/2 Second case: √m = (9 - 3√5)/2 (√m)² = [(9 - 3√5)/2]² m = (9 - 3√5)²/4 m = (81 - 54√5 + 45)/4 m = (126 - 54√5)/4 m = (63 - 27√5)/2 m = 9.(7 - 3√5)/2 → then you can deduce that: n = 9.(7 + 3√5)/2
√m + √n = 9 ← this is the sum S
√m * √n = 9 ← this is the product P
√m & √n are the solution of the following equation:
x² - Sx + P = 0 → where S is the sum and where P is the product
x² - 9x + 9 = 0
Δ = (- 9)² - (4 * 9) = 81 - 36 = 45
x = (9 ± √45)/2
x = (9 ± 3√5)/2 ← these are √m & √n
First case: √m = (9 + 3√5)/2
(√m)² = [(9 + 3√5)/2]²
m = (9 + 3√5)²/4
m = (81 + 54√5 + 45)/4
m = (126 + 54√5)/4
m = (63 + 27√5)/2
m = 9.(7 + 3√5)/2 → then you can deduce that:
n = 9.(7 - 3√5)/2
Second case: √m = (9 - 3√5)/2
(√m)² = [(9 - 3√5)/2]²
m = (9 - 3√5)²/4
m = (81 - 54√5 + 45)/4
m = (126 - 54√5)/4
m = (63 - 27√5)/2
m = 9.(7 - 3√5)/2 → then you can deduce that:
n = 9.(7 + 3√5)/2
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