Since you have found both the Product and the Sum of a,b you can say that a,b are the roots of t^2-S*t+P=0. Or you can avoid finding the Product at all and just substitute directly the a= -1-b (or b=-1-a) on any of given equations, a^2=b+183==>a^2=-a-1+183a^2+a-182=0 (or b^2+b-182=0). So many unnecessary steps are more tiring than helpful....
This is very easy: if a=b, then a^2-a-183=0. Solve using the formula for quadratic equations to obtain two solutions a=(1+√733)/2 and a=(1-√733)/2. If a not equal b: subtract the second equation from the first to obtain a*2-b^2=b-a. Divide both sides by (a-b): a+b=-1, therefore b=-(a+1). Replace b by -(a+1) in first equation: a^2+a-182=0. Solve using formula for quadratic equation ==> a=13, b=-14 or a=-14, b=13. As shown in the clip, more convoluted solutions are possible too...
(1): a² = b + 183 (2): b² = a + 183 (1) - (2) a² - b² = (b + 183) - (a + 183) (a + b).(a - b) = b + 183 - a - 183 (a + b).(a - b) = b - a (a + b).(a - b) - (b - a) = 0 (a + b).(a - b) + (a - b) = 0 (a - b).[(a + b) + 1] = 0 → where: a ≠ b → (a - b) ≠ 0 a + b + 1 = 0 a + b = - 1 ← equation (3) (1) + (2) a² + b² = (b + 183) + (a + 183) a² + b² = b + a + 366 → recall (3): a + b = - 1 a² + b² = 365 ← equation (4) From (3): a + b = - 1 (a + b)² = 1 a² + b² + 2ab = 1 → recall (4): a² + b² = 365 365 + 2ab = 1 2ab = - 364 ← equation (5) (a - b)² = a² + b² - 2ab → recall (4): a² + b² = 365 (a - b)² = 365 - 2ab → recall (5): 2ab = - 364 (a - b)² = 365 + 364 (a - b)² = 729 a - b = ± 27 ← equation (6) First case: a - b = 27 a - b = 27 → recall (3): a + b = - 1 a + b = - 1 --------------------------------------------Sum 2a = 26 → a = 13 Second case: a - b = - 27 a - b = - 27 → recall (3): a + b = - 1 a + b = - 1 --------------------------------------------Sum 2a = - 28 → a = - 14
Since you have found both the Product and the Sum of a,b you can say that a,b are the roots of t^2-S*t+P=0. Or you can avoid finding the Product at all and just substitute directly the a= -1-b (or b=-1-a) on any of given equations, a^2=b+183==>a^2=-a-1+183a^2+a-182=0 (or b^2+b-182=0). So many unnecessary steps are more tiring than helpful....
Such a nice work... good job
Much appreciated!👏👏👏✅✅✅
This is very easy: if a=b, then a^2-a-183=0. Solve using the formula for quadratic equations to obtain two solutions a=(1+√733)/2 and a=(1-√733)/2. If a not equal b: subtract the second equation from the first to obtain
a*2-b^2=b-a. Divide both sides by (a-b):
a+b=-1, therefore b=-(a+1). Replace b by -(a+1) in first equation:
a^2+a-182=0. Solve using formula for quadratic equation ==> a=13, b=-14 or a=-14, b=13.
As shown in the clip, more convoluted solutions are possible too...
a^2=b+183
b^2=a+183
a=奇數，b=偶數
a^2=1mod8
b^2
=4 or8mod8
a^2=b+183
1mod8
=-(6mod8)+7mod8
b=-(6mod8)
b=-14
a^2=-14+183
a^2=169
a=+-(13), b=-14
b=+-(13), a=-14
a=+-(14), b=13
b=+-(14), a=13
(1): a² = b + 183
(2): b² = a + 183
(1) - (2)
a² - b² = (b + 183) - (a + 183)
(a + b).(a - b) = b + 183 - a - 183
(a + b).(a - b) = b - a
(a + b).(a - b) - (b - a) = 0
(a + b).(a - b) + (a - b) = 0
(a - b).[(a + b) + 1] = 0 → where: a ≠ b → (a - b) ≠ 0
a + b + 1 = 0
a + b = - 1 ← equation (3)
(1) + (2)
a² + b² = (b + 183) + (a + 183)
a² + b² = b + a + 366 → recall (3): a + b = - 1
a² + b² = 365 ← equation (4)
From (3):
a + b = - 1
(a + b)² = 1
a² + b² + 2ab = 1 → recall (4): a² + b² = 365
365 + 2ab = 1
2ab = - 364 ← equation (5)
(a - b)² = a² + b² - 2ab → recall (4): a² + b² = 365
(a - b)² = 365 - 2ab → recall (5): 2ab = - 364
(a - b)² = 365 + 364
(a - b)² = 729
a - b = ± 27 ← equation (6)
First case: a - b = 27
a - b = 27 → recall (3): a + b = - 1
a + b = - 1
--------------------------------------------Sum
2a = 26
→ a = 13
Second case: a - b = - 27
a - b = - 27 → recall (3): a + b = - 1
a + b = - 1
--------------------------------------------Sum
2a = - 28
→ a = - 14