Question

If $a-b=3$ and $a^3-b^3=117$, then find the value of $a+b$?

Answer 1

Find the value of $a+b$

Given that

1. $a-b=3$
2. $a^3-b^3=117$

Using identity, $a^3-b^3=(a-b)(a^2+ab+b^2)$

$$\begin{gathered} \Rightarrow 117=\left(3\right)[{(a-b)}^2+3ab]\mathbf{[}{\mathit{a}}^{\mathbf{2}}\mathbf{+}{\mathit{b}}^{\mathbf{2}}\mathbf{=}{\mathbf{(}\mathit{a}\mathbf{-}\mathit{b}\mathbf{)}}^{\mathbf{2}}\mathbf{+}\mathbf{2}\mathit{a}\mathit{b}\mathbf{]} \\ \Rightarrow 117=3[{\left(3\right)}^2+3ab] \\ \Rightarrow 39=9+3ab \\ \Rightarrow 30=3ab \\ \Rightarrow 10=ab \end{gathered}$$

Now,

${(a+b)}^2={(a-b)}^2+4ab$

$$\begin{gathered} \Rightarrow {(a+b)}^2={\left(3\right)}^2+4\times 10 \\ \Rightarrow {(a+b)}^2=9+40 \\ \Rightarrow {(a+b)}^2=49 \\ \Rightarrow a+b=7 \end{gathered}$$

Hence, $a+b$ is equal to $7$