Solve 4x^2-5xy+5xy-81y^2/9=(2x-9y/3)(2x-9y/3)

$$\frac { 4 x ^ { 2 } - 5 x y + 5 x y - 81 y ^ { 2 } } { 9 } = ( \frac { 2 x - 9 y } { 3 } ) ( \frac { 2 x - 9 y } { 3 } )$$

Solve for x (complex solution)

$$\left\{\begin{matrix}\\x=\frac{9y}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{C}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 9, the least common multiple of 9,3.

$$4x^{2}-5xy+5xy-81y^{2}=\left(2x-9y\right)\left(2x-9y\right)$$

Multiply 2x-9y and 2x-9y to get \left(2x-9y\right)^{2}.

$$4x^{2}-5xy+5xy-81y^{2}=\left(2x-9y\right)^{2}$$

Combine -5xy and 5xy to get 0.

$$4x^{2}-81y^{2}=\left(2x-9y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-9y\right)^{2}.

$$4x^{2}-81y^{2}=4x^{2}-36xy+81y^{2}$$

Subtract 4x^{2} from both sides.

$$4x^{2}-81y^{2}-4x^{2}=-36xy+81y^{2}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-81y^{2}=-36xy+81y^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-36xy+81y^{2}=-81y^{2}$$

Subtract 81y^{2} from both sides.

$$-36xy=-81y^{2}-81y^{2}$$

Combine -81y^{2} and -81y^{2} to get -162y^{2}.

$$-36xy=-162y^{2}$$

The equation is in standard form.

$$\left(-36y\right)x=-162y^{2}$$

Divide both sides by -36y.

$$\frac{\left(-36y\right)x}{-36y}=-\frac{162y^{2}}{-36y}$$

Dividing by -36y undoes the multiplication by -36y.

$$x=-\frac{162y^{2}}{-36y}$$

Divide -162y^{2} by -36y.

$$x=\frac{9y}{2}$$

Solve for x

$$\left\{\begin{matrix}\\x=\frac{9y}{2}\text{, }&\text{unconditionally}\\x\in \mathrm{R}\text{, }&y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 9, the least common multiple of 9,3.

$$4x^{2}-5xy+5xy-81y^{2}=\left(2x-9y\right)\left(2x-9y\right)$$

Multiply 2x-9y and 2x-9y to get \left(2x-9y\right)^{2}.

$$4x^{2}-5xy+5xy-81y^{2}=\left(2x-9y\right)^{2}$$

Combine -5xy and 5xy to get 0.

$$4x^{2}-81y^{2}=\left(2x-9y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2x-9y\right)^{2}.

$$4x^{2}-81y^{2}=4x^{2}-36xy+81y^{2}$$

Subtract 4x^{2} from both sides.

$$4x^{2}-81y^{2}-4x^{2}=-36xy+81y^{2}$$

Combine 4x^{2} and -4x^{2} to get 0.

$$-81y^{2}=-36xy+81y^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$-36xy+81y^{2}=-81y^{2}$$

Subtract 81y^{2} from both sides.

$$-36xy=-81y^{2}-81y^{2}$$

Combine -81y^{2} and -81y^{2} to get -162y^{2}.

$$-36xy=-162y^{2}$$

The equation is in standard form.

$$\left(-36y\right)x=-162y^{2}$$

Divide both sides by -36y.

$$\frac{\left(-36y\right)x}{-36y}=-\frac{162y^{2}}{-36y}$$

Dividing by -36y undoes the multiplication by -36y.

$$x=-\frac{162y^{2}}{-36y}$$

Divide -162y^{2} by -36y.

$$x=\frac{9y}{2}$$

Solve for y

$$y=\frac{2x}{9}$$

$$y=0$$