How can I solve this factorization problem without assumptions?

[Milind_shyani]

Hello,
I have got one mathematical problem and i am not able to solve it may i plese get some help. My sum is as foolws:-
a^2+b^2=25 and a^3+b^3=91 so now find the values of a and b.
Now here we cannot tahe into consideration that as a^2+b^2=25 , a=3 and b=4 or a=4 and b=3 .But we have to find it by mathematical formulas and without any assumptions. thank you in advance. Please send it soon

 

[finchie_88]

$$a^2 + b^2 = 25 \therefore a = \sqrt{25 - b^2}$$
$$a = (91 - b^3)^{1/3}$$
Combine the two and solve to get a value (possibly more than one value), and then use these to work out value(s) for a. The re-arranging might be a little complex, but it should be do-able.

 

[VietDao29]

Hello,
I have got one mathematical problem and i am not able to solve it may i plese get some help. My sum is as foolws:-
a^2+b^2=25 and a^3+b^3=91 so now find the values of a and b.
Now here we cannot tahe into consideration that as a^2+b^2=25 , a=3 and b=4 or a=4 and b=3 .But we have to find it by mathematical formulas and without any assumptions. thank you in advance. Please send it soon

$$\left\{ \begin{array}{l} a ^ 2 + b ^ 2 = 25 \quad (1) \\ a ^ 3 + b ^ 3 = 91 \quad (2) \end{array} \right.$$
I think you may want to try this:
Now let x = a + b, and y = ab, we will try to write the equation (1), and (2) in terms of x, and y:
a² + b² = (a + b)² - 2ab = x² - 2y
a³ + b³ = (a + b) (a² - ab + b²) = (a + b) ((a + b)² - 3ab) = x (x² - 3y) = x³ - 3xy.
So you'll have:
$$\left\{ \begin{array}{l} x ^ 2 - 2y = 25 \quad (3) \\ x ^ 3 - 3xy = 91 \quad (4) \end{array} \right.$$
Now from the equation (3), one can solve y in terms of x, then plug y in equation (4), and solve for x. From there, you can solve for y.
having x = a + b, and y = ab, one then can find a, and b.
Can you go from here? :)