Question

What must be subtracted from $\frac{-2}{3}{x}^2+\frac{3}{2}xy+2y^2$ to get $3x^2-2xy-\frac{7}{2}y$.

Answer 1

Let $z$ be the expression being subtracted.

$-\frac{2}{3}{x}^2+\frac{3}{2}xy+2y^2-z=3x^2-2xy-\frac{7}{2}y$

$\Rightarrow z=-\frac{2}{3}{x}^2+\frac{3}{2}xy+2y^2-(3x^2-2xy-\frac{7}{2}y)$

$\Rightarrow z=-\frac{2}{3}{x}^2+\frac{3}{2}xy+2y^2-3x^2+2xy+\frac{7}{2}y$

$\Rightarrow z=\left(-\frac{2}{3}-3\right)x^2+\left(\frac{3}{2}+2\right)xy+2y^2+\frac{7}{2}y$

$\Rightarrow z=-\frac{11}{3}{x}^2+\frac{7}{2}xy+2y^2+\frac{7}{2}y$

Hence, the expression to be subtracted is $-\frac{11}{3}{x}^2+\frac{7}{2}xy+2y^2+\frac{7}{2}y$.