Solve w=2(0-2x)(6-2x)+0.5[2x(10-2x)+2x(6-2x)]

$$w = 2 ( 0 - 2 x ) ( 6 - 2 x ) + 0.5 [ 2 x ( 10 - 2 x ) + 2 x ( 6 - 2 x ) ]$$

Solve for w

$$w=8x\left(\frac{x}{2}-1\right)$$

Solution Steps

Anything plus zero gives itself.

$$w=2\left(-2\right)x\left(6-2x\right)+0.5\left(2x\left(10-2x\right)+2x\left(6-2x\right)\right)$$

Multiply 2 and -2 to get -4.

$$w=-4x\left(6-2x\right)+0.5\left(2x\left(10-2x\right)+2x\left(6-2x\right)\right)$$

Use the distributive property to multiply -4x by 6-2x.

$$w=-24x+8x^{2}+0.5\left(2x\left(10-2x\right)+2x\left(6-2x\right)\right)$$

Use the distributive property to multiply 2x by 10-2x.

$$w=-24x+8x^{2}+0.5\left(20x-4x^{2}+2x\left(6-2x\right)\right)$$

Use the distributive property to multiply 2x by 6-2x.

$$w=-24x+8x^{2}+0.5\left(20x-4x^{2}+12x-4x^{2}\right)$$

Combine 20x and 12x to get 32x.

$$w=-24x+8x^{2}+0.5\left(32x-4x^{2}-4x^{2}\right)$$

Combine -4x^{2} and -4x^{2} to get -8x^{2}.

$$w=-24x+8x^{2}+0.5\left(32x-8x^{2}\right)$$

Use the distributive property to multiply 0.5 by 32x-8x^{2}.

$$w=-24x+8x^{2}+16x-4x^{2}$$

Combine -24x and 16x to get -8x.

$$w=-8x+8x^{2}-4x^{2}$$

Combine 8x^{2} and -4x^{2} to get 4x^{2}.

$$w=-8x+4x^{2}$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{w+4}}{2}+1$$

$$x=\frac{\sqrt{w+4}}{2}+1$$

Solve for x

$$x=-\frac{\sqrt{w+4}}{2}+1$$

$$x=\frac{\sqrt{w+4}}{2}+1\text{, }w\geq -4$$