Solve {l}{(y-3)^2-2(y-2)^2}{(2a-3)^2-3q(q-4)}

$$\left. \begin{array} { l } { ( y - 3 ) ^ { 2 } - 2 ( y - 2 ) ^ { 2 } } \\ { ( 2 a - 3 ) ^ { 2 } - 3 q ( q - 4 ) } \end{array} \right.$$

Least Common Multiple

$$3\left(qy\right)^{2}-4\left(ay\right)^{2}+8ya^{2}+12ay^{2}-6yq^{2}-12qy^{2}+24qy-24ay+\left(2a-3\right)^{2}-3q^{2}-9y^{2}+12q+18y$$

Solution Steps

Factor the expressions that are not already factored.

$$1+2y-y^{2}=-\left(y-\left(\sqrt{2}+1\right)\right)\left(y-\left(-\sqrt{2}+1\right)\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$3q^{2}y^{2}-4a^{2}y^{2}+8ya^{2}+12ay^{2}-6yq^{2}-12qy^{2}+24qy-24ay+\left(2a-3\right)^{2}-3q^{2}-9y^{2}+12q+18y$$

Expand the expression.

$$3q^{2}y^{2}-4a^{2}y^{2}+12ay^{2}-12qy^{2}-9y^{2}+8ya^{2}-6yq^{2}+24qy-24ay+18y-3q^{2}+4a^{2}-12a+12q+9$$

Evaluate

$$1+2y-y^{2},\ -3q\left(q-4\right)+\left(2a-3\right)^{2}$$