Question

If $x+y=a$ and $x^2+y^2=b$, then the value of $(x^3+y^3)$, is

• A. ab
• B. $a^2+b$
• C. $a+b^2$
• D. $\frac{3ab-a^3}{2}$

Answer 1

The correct option is D $\frac{3ab-a^3}{2}$

We have,

$x+y=a......\left(1\right)$

$x^2+y^2=b......\left(2\right)$

From equation (1) to, and we get,

${(x+y)}^2=a^2$

$\Rightarrow x^2+y^2+2xy=a^2$

$\Rightarrow b+2xy=a^2$

$\Rightarrow xy=\frac{a^2-b}{2}$

Using formula $A^3+B^3=(A+B)(A^2+B^2-AB)$

Now,

$x^3+y^3=(x+y)(x^2+y^2-xy)$

$=a[b-\left(\frac{a^2-b}{2}\right)]$

$=a\left[\frac{2b-a^2+b}{2}\right]$

$=\left(\frac{3ab-a^3}{2}\right)$

Hence, this is the answer.