Question

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

• A. $2x^2+2z^2$
• B. $2y^2$
• C. $2x^2+2y^2-z^2$
• D. $2y^2-4z^2$

Answer 1

Answer: (D)

$2y^2-4z^2$

Add two expressions

$\Rightarrow (4x^2-5y^2+3z^2)+(-3x^2+4y^2+2z^2)$

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

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

$=x^2-y^2+5z^2$

But $x^2-y^2+5z^2+P\left(x\right)=x^2+y^2+z^2$, where $P\left(x\right)$ is the third expression

$\therefore P\left(x\right)=(x^2+y^2+z^2)-(x^2-y^2+5z^2)$

$=x^2+y^2+z^2-x^2+y^2-5z^2=2y^2-4z^2$