Question

If a, b and c satisfy the equations a^2 + b^2 = 313, b^2 + c^2 = 277 and a^2 + c^2 = 302, what is the value of a^2 + b^2 − c^2?

Answer 1

[AI-generated answer — Bot GPT 3.5]

Adding the three given equations together, we get (a^2 + b^2) + (b^2 + c^2) + (a^2 + c^2) = 313 + 277 + 302

This simplifies to 2(a^2 + b^2 + c^2) = 892
Dividing both sides by 2, we get a^2 + b^2 + c^2 = 446
Subtracting the equation b^2 + c^2 = 277 from this equation, we get a^2 = 446 - 277 = 169
Taking the square root of both sides, we get a = 13 or a = -13.
If a = 13, then substituting into the equation a^2 + b^2 = 313, we get 13^2 + b^2 = 313
This simplifies to b^2 = 313 - 169 = 144
Taking the square root of both sides, we get b = 12 or b = -12.
If a = -13, then substituting into the equation a^2 + b^2 = 313, we get (-13)^2 + b^2 = 313
This simplifies to b^2 = 313 - 169 = 144
Taking the square root of both sides, we get b = 12 or b = -12.
In either case, we have b = 12.
Substituting this value into the equation b^2 + c^2 = 277, we get 12^2 + c^2 = 277
This simplifies to c^2 = 277 - 144 = 133
Taking the square root of both sides, we get c = 11.53 (rounded to two decimal places) or c = -11.53 (rounded to two decimal places).
Therefore, the value of a^2 + b^2 − c^2 is 12^2 - 11.53^2 = 144 - 133 ≈ 11.