Solve (x/2-t)^2(x/2+t)^2-(x/2-t)(x/2+t)(frac{x^2}{4}+t^2)=-frac{1}{2}(1-tx)(2-tx)

$$( \frac { x } { 2 } - t ) ^ { 2 } ( \frac { x } { 2 } + t ) ^ { 2 } - ( \frac { x } { 2 } - t ) ( \frac { x } { 2 } + t ) ( \frac { x ^ { 2 } } { 4 } + t ^ { 2 } ) = - \frac { 1 } { 2 } ( 1 - t x ) ( 2 - t x )$$

Solve for x

$$x=\frac{4t^{3}}{3}+\frac{2}{3t}$$

$$t\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$4\left(\frac{x}{2}-t\right)^{2}\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

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

$$4\left(\left(\frac{x}{2}\right)^{2}-2\times \frac{x}{2}t+t^{2}\right)\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

$$4\left(\frac{x^{2}}{2^{2}}-2\times \frac{x}{2}t+t^{2}\right)\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Express -2\times \frac{x}{2} as a single fraction.

$$4\left(\frac{x^{2}}{2^{2}}+\frac{-2x}{2}t+t^{2}\right)\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Cancel out 2 and 2.

$$4\left(\frac{x^{2}}{2^{2}}-xt+t^{2}\right)\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -xt+t^{2} times \frac{2^{2}}{2^{2}}.

$$4\left(\frac{x^{2}}{2^{2}}+\frac{\left(-xt+t^{2}\right)\times 2^{2}}{2^{2}}\right)\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Since \frac{x^{2}}{2^{2}} and \frac{\left(-xt+t^{2}\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$4\times \frac{x^{2}+\left(-xt+t^{2}\right)\times 2^{2}}{2^{2}}\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Do the multiplications in x^{2}+\left(-xt+t^{2}\right)\times 2^{2}.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\left(\frac{x}{2}+t\right)^{2}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

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

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\left(\left(\frac{x}{2}\right)^{2}+2\times \frac{x}{2}t+t^{2}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

To raise \frac{x}{2} to a power, raise both numerator and denominator to the power and then divide.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\left(\frac{x^{2}}{2^{2}}+2\times \frac{x}{2}t+t^{2}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Express 2\times \frac{x}{2} as a single fraction.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\left(\frac{x^{2}}{2^{2}}+\frac{2x}{2}t+t^{2}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Cancel out 2 and 2.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\left(\frac{x^{2}}{2^{2}}+xt+t^{2}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

To add or subtract expressions, expand them to make their denominators the same. Multiply xt+t^{2} times \frac{2^{2}}{2^{2}}.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\left(\frac{x^{2}}{2^{2}}+\frac{\left(xt+t^{2}\right)\times 2^{2}}{2^{2}}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Since \frac{x^{2}}{2^{2}} and \frac{\left(xt+t^{2}\right)\times 2^{2}}{2^{2}} have the same denominator, add them by adding their numerators.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\times \frac{x^{2}+\left(xt+t^{2}\right)\times 2^{2}}{2^{2}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Do the multiplications in x^{2}+\left(xt+t^{2}\right)\times 2^{2}.

$$4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}}\times \frac{x^{2}+4xt+4t^{2}}{2^{2}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Express 4\times \frac{x^{2}-4xt+4t^{2}}{2^{2}} as a single fraction.

$$\frac{4\left(x^{2}-4xt+4t^{2}\right)}{2^{2}}\times \frac{x^{2}+4xt+4t^{2}}{2^{2}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Multiply \frac{4\left(x^{2}-4xt+4t^{2}\right)}{2^{2}} times \frac{x^{2}+4xt+4t^{2}}{2^{2}} by multiplying numerator times numerator and denominator times denominator.

$$\frac{4\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)}{2^{2}\times 2^{2}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-2\left(1-tx\right)\left(2-tx\right)$$

Use the distributive property to multiply -2 by 1-tx.

$$\frac{4\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)}{2^{2}\times 2^{2}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=\left(-2+2tx\right)\left(2-tx\right)$$

Use the distributive property to multiply -2+2tx by 2-tx and combine like terms.

$$\frac{4\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)}{2^{2}\times 2^{2}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-4+6tx-2t^{2}x^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$\frac{4\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)}{2^{4}}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-4+6tx-2t^{2}x^{2}$$

Calculate 2 to the power of 4 and get 16.

$$\frac{4\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)}{16}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-4+6tx-2t^{2}x^{2}$$

Divide 4\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right) by 16 to get \frac{1}{4}\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right).

$$\frac{1}{4}\left(x^{2}-4xt+4t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-4+6tx-2t^{2}x^{2}$$

Use the distributive property to multiply \frac{1}{4} by x^{2}-4xt+4t^{2}.

$$\left(\frac{1}{4}x^{2}-xt+t^{2}\right)\left(x^{2}+4xt+4t^{2}\right)-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-4+6tx-2t^{2}x^{2}$$

Use the distributive property to multiply \frac{1}{4}x^{2}-xt+t^{2} by x^{2}+4xt+4t^{2} and combine like terms.

$$\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)=-4+6tx-2t^{2}x^{2}$$

Subtract 6tx from both sides.

$$\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)-6tx=-4-2t^{2}x^{2}$$

Add 2t^{2}x^{2} to both sides.

$$\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)-6tx+2t^{2}x^{2}=-4$$

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$4\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-24tx+8t^{2}x^{2}=-16$$

Multiply both sides of the equation by 4, the least common multiple of 4,2.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-4\left(\frac{x}{2}-t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Use the distributive property to multiply -4 by \frac{x}{2}-t.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(-4\times \frac{x}{2}+4t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Cancel out 2, the greatest common factor in 4 and 2.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(-2x+4t\right)\left(\frac{x}{2}+t\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Use the distributive property to multiply -2x+4t by \frac{x}{2}+t.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(-2x\times \frac{x}{2}-2xt+4t\times \frac{x}{2}+4t^{2}\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Express -2\times \frac{x}{2} as a single fraction.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(\frac{-2x}{2}x-2xt+4t\times \frac{x}{2}+4t^{2}\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Cancel out 2 and 2.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(-xx-2xt+4t\times \frac{x}{2}+4t^{2}\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Cancel out 2, the greatest common factor in 4 and 2.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(-xx-2xt+2xt+4t^{2}\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Combine -2xt and 2xt to get 0.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}+\left(-xx+4t^{2}\right)\left(\frac{x^{2}}{4}+t^{2}\right)\right)-96tx+32t^{2}x^{2}=-64$$

Use the distributive property to multiply -xx+4t^{2} by \frac{x^{2}}{4}+t^{2}.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-\frac{x^{2}}{4}x^{2}-t^{2}x^{2}+4t^{2}\times \frac{x^{2}}{4}+4t^{4}\right)-96tx+32t^{2}x^{2}=-64$$

Express \frac{x^{2}}{4}x^{2} as a single fraction.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-\frac{x^{2}x^{2}}{4}-t^{2}x^{2}+4t^{2}\times \frac{x^{2}}{4}+4t^{4}\right)-96tx+32t^{2}x^{2}=-64$$

Express 4\times \frac{x^{2}}{4} as a single fraction.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-\frac{x^{2}x^{2}}{4}-t^{2}x^{2}+\frac{4x^{2}}{4}t^{2}+4t^{4}\right)-96tx+32t^{2}x^{2}=-64$$

Cancel out 4 and 4.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-\frac{x^{2}x^{2}}{4}-t^{2}x^{2}+x^{2}t^{2}+4t^{4}\right)-96tx+32t^{2}x^{2}=-64$$

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

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-\frac{x^{2}x^{2}}{4}+4t^{4}\right)-96tx+32t^{2}x^{2}=-64$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+4t^{4}-\frac{x^{4}}{4}+4t^{4}\right)-96tx+32t^{2}x^{2}=-64$$

Combine 4t^{4} and 4t^{4} to get 8t^{4}.

$$16\left(\frac{1}{4}x^{4}-2x^{2}t^{2}+8t^{4}-\frac{x^{4}}{4}\right)-96tx+32t^{2}x^{2}=-64$$

Use the distributive property to multiply 16 by \frac{1}{4}x^{4}-2x^{2}t^{2}+8t^{4}-\frac{x^{4}}{4}.

$$4x^{4}-32x^{2}t^{2}+128t^{4}-16\times \frac{x^{4}}{4}-96tx+32t^{2}x^{2}=-64$$

Cancel out 4, the greatest common factor in 16 and 4.

$$4x^{4}-32x^{2}t^{2}+128t^{4}-4x^{4}-96tx+32t^{2}x^{2}=-64$$

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

$$-32x^{2}t^{2}+128t^{4}-96tx+32t^{2}x^{2}=-64$$

Combine -32x^{2}t^{2} and 32t^{2}x^{2} to get 0.

$$128t^{4}-96tx=-64$$

Subtract 128t^{4} from both sides.

$$-96tx=-64-128t^{4}$$

The equation is in standard form.

$$\left(-96t\right)x=-128t^{4}-64$$

Divide both sides by -96t.

$$\frac{\left(-96t\right)x}{-96t}=\frac{-128t^{4}-64}{-96t}$$

Dividing by -96t undoes the multiplication by -96t.

$$x=\frac{-128t^{4}-64}{-96t}$$

Divide -64-128t^{4} by -96t.

$$x=\frac{4t^{3}}{3}+\frac{2}{3t}$$