Quadratic equations with $a$ and $b$

Question

If $a≠b$, and $a²=5a-7$ and $b²=5b-7$, then $a³+b³=?$ How can I find the answer? I found out that $a$ and $b$ are not real numbers. Can anyone please teach me? thank you!!

Answer 1

You can do it in this way:

$a^3+b^3=(a+b)^3-3ab(a+b)$

Easily you can know that $a,b$ are two solutions of equation $x^2-5x+7=0$, so you can know that $a+b=5$ and $ab=7$, so you can get

$a^3+b^3=5^3-3\times7\times5=20$

Answer 2

An alternative solution, without Vieta's formulas or binomial expansion or sum of cubes factors.

First subtract the two equations to get $a^2-b^2 = 5(a-b)$. Since it's given that $a \ne b\,$, divide by $a-b$ which leaves $a+b=5$.

Next note that $a^2=5a-7\,$, so $\,a^3= 5a^2-7a=5(5a-7)-7a=18a -35\,$.

Same works for $b$, then adding the two: $\,a^3+b^3=18(a+b)-70=18 \cdot 5 - 70 = 20$.