Question

The sum of three expressions is $x^2+y^2+z^2.$ If two of them are $4x^2-5y^2+z^2$ and $-3x^2+4y^2+2z^2$, find the third expression.

• A. $2x^2+2z^2$
• B. $2y^2$
• C. $2x^2+2y^2-z^2$
• D. $2y^2-2z^2$

Answer 1

The correct option is D $2y^2-2z^2$

Let the third expression be $a$ then,

$a+(4x^2-5y^2+z^2)+(-3x^2+4y^2+2z^2)=x^2+y^2+z^2$

$\Rightarrow a+(x^2-y^2+3z^2)=(x^2+y^2+z^2)$

$\Rightarrow a=(x^2+y^2+z^2)-x^2+y^2-3z^2$

$\Rightarrow a=2y^2-2z^2$

Hence, option $D$ is correct.