Solve 16x^2-36y^2=(4x+6y)(4x-6y)

$$16 x ^ { 2 } - 36 y ^ { 2 } = ( 4 x + 6 y ) ( 4 x - 6 y )$$

Solve for x (complex solution)

$$x\in \mathrm{C}$$

Solution Steps

Consider \left(4x+6y\right)\left(4x-6y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$16x^{2}-36y^{2}=\left(4x\right)^{2}-\left(6y\right)^{2}$$

Expand \left(4x\right)^{2}.

$$16x^{2}-36y^{2}=4^{2}x^{2}-\left(6y\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-36y^{2}=16x^{2}-\left(6y\right)^{2}$$

Expand \left(6y\right)^{2}.

$$16x^{2}-36y^{2}=16x^{2}-6^{2}y^{2}$$

Calculate 6 to the power of 2 and get 36.

$$16x^{2}-36y^{2}=16x^{2}-36y^{2}$$

Subtract 16x^{2} from both sides.

$$16x^{2}-36y^{2}-16x^{2}=-36y^{2}$$

Combine 16x^{2} and -16x^{2} to get 0.

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

Cancel out -36 on both sides.

$$y^{2}=y^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{C}$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

Solution Steps

Consider \left(4x+6y\right)\left(4x-6y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$16x^{2}-36y^{2}=\left(4x\right)^{2}-\left(6y\right)^{2}$$

Expand \left(4x\right)^{2}.

$$16x^{2}-36y^{2}=4^{2}x^{2}-\left(6y\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-36y^{2}=16x^{2}-\left(6y\right)^{2}$$

Expand \left(6y\right)^{2}.

$$16x^{2}-36y^{2}=16x^{2}-6^{2}y^{2}$$

Calculate 6 to the power of 2 and get 36.

$$16x^{2}-36y^{2}=16x^{2}-36y^{2}$$

Add 36y^{2} to both sides.

$$16x^{2}-36y^{2}+36y^{2}=16x^{2}$$

Combine -36y^{2} and 36y^{2} to get 0.

$$16x^{2}=16x^{2}$$

Cancel out 16 on both sides.

$$x^{2}=x^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{C}$$

Solve for x

$$x\in \mathrm{R}$$

Solution Steps

Consider \left(4x+6y\right)\left(4x-6y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$16x^{2}-36y^{2}=\left(4x\right)^{2}-\left(6y\right)^{2}$$

Expand \left(4x\right)^{2}.

$$16x^{2}-36y^{2}=4^{2}x^{2}-\left(6y\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-36y^{2}=16x^{2}-\left(6y\right)^{2}$$

Expand \left(6y\right)^{2}.

$$16x^{2}-36y^{2}=16x^{2}-6^{2}y^{2}$$

Calculate 6 to the power of 2 and get 36.

$$16x^{2}-36y^{2}=16x^{2}-36y^{2}$$

Subtract 16x^{2} from both sides.

$$16x^{2}-36y^{2}-16x^{2}=-36y^{2}$$

Combine 16x^{2} and -16x^{2} to get 0.

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

Cancel out -36 on both sides.

$$y^{2}=y^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any x.

$$x\in \mathrm{R}$$

Solve for y

$$y\in \mathrm{R}$$

Solution Steps

Consider \left(4x+6y\right)\left(4x-6y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$16x^{2}-36y^{2}=\left(4x\right)^{2}-\left(6y\right)^{2}$$

Expand \left(4x\right)^{2}.

$$16x^{2}-36y^{2}=4^{2}x^{2}-\left(6y\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$16x^{2}-36y^{2}=16x^{2}-\left(6y\right)^{2}$$

Expand \left(6y\right)^{2}.

$$16x^{2}-36y^{2}=16x^{2}-6^{2}y^{2}$$

Calculate 6 to the power of 2 and get 36.

$$16x^{2}-36y^{2}=16x^{2}-36y^{2}$$

Add 36y^{2} to both sides.

$$16x^{2}-36y^{2}+36y^{2}=16x^{2}$$

Combine -36y^{2} and 36y^{2} to get 0.

$$16x^{2}=16x^{2}$$

Cancel out 16 on both sides.

$$x^{2}=x^{2}$$

Reorder the terms.

$$\text{true}$$

This is true for any y.

$$y\in \mathrm{R}$$