Solve 17x-34x^2/9+(1/2-2x)^2-1/24=x^2(x-1/2)-x(x-frac{1}{3})^2

$$\frac { 17 x - 34 x ^ { 2 } } { 9 } + ( \frac { 1 } { 2 } - 2 x ) ^ { 2 } - \frac { 1 } { 24 } = x ^ { 2 } ( x - \frac { 1 } { 2 } ) - x ( x - \frac { 1 } { 3 } ) ^ { 2 }$$

Solve for x (complex solution)

$$x=-\frac{\sqrt{15}i}{2}\approx -0-1.936491673i$$

$$x=\frac{\sqrt{15}i}{2}\approx 1.936491673i$$

Steps by Finding Square Root

Multiply both sides of the equation by 72, the least common multiple of 9,24,2.

$$8\left(17x-34x^{2}\right)+72\left(\frac{1}{2}-2x\right)^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use the distributive property to multiply 8 by 17x-34x^{2}.

$$136x-272x^{2}+72\left(\frac{1}{2}-2x\right)^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{2}-2x\right)^{2}.

$$136x-272x^{2}+72\left(\frac{1}{4}-2x+4x^{2}\right)-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use the distributive property to multiply 72 by \frac{1}{4}-2x+4x^{2}.

$$136x-272x^{2}+18-144x+288x^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Combine 136x and -144x to get -8x.

$$-8x-272x^{2}+18+288x^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Combine -272x^{2} and 288x^{2} to get 16x^{2}.

$$-8x+16x^{2}+18-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Subtract 3 from 18 to get 15.

$$-8x+16x^{2}+15=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use the distributive property to multiply 72x^{2} by x-\frac{1}{2}.

$$-8x+16x^{2}+15=72x^{3}-36x^{2}-72x\left(x-\frac{1}{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{3}\right)^{2}.

$$-8x+16x^{2}+15=72x^{3}-36x^{2}-72x\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right)$$

Add 72x\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right) to both sides.

$$-8x+16x^{2}+15+72x\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right)=72x^{3}-36x^{2}$$

Use the distributive property to multiply 72x by x^{2}-\frac{2}{3}x+\frac{1}{9}.

$$-8x+16x^{2}+15+72x^{3}-48x^{2}+8x=72x^{3}-36x^{2}$$

Combine 16x^{2} and -48x^{2} to get -32x^{2}.

$$-8x-32x^{2}+15+72x^{3}+8x=72x^{3}-36x^{2}$$

Combine -8x and 8x to get 0.

$$-32x^{2}+15+72x^{3}=72x^{3}-36x^{2}$$

Subtract 72x^{3} from both sides.

$$-32x^{2}+15+72x^{3}-72x^{3}=-36x^{2}$$

Combine 72x^{3} and -72x^{3} to get 0.

$$-32x^{2}+15=-36x^{2}$$

Add 36x^{2} to both sides.

$$-32x^{2}+15+36x^{2}=0$$

Combine -32x^{2} and 36x^{2} to get 4x^{2}.

$$4x^{2}+15=0$$

Subtract 15 from both sides. Anything subtracted from zero gives its negation.

$$4x^{2}=-15$$

Divide both sides by 4.

$$x^{2}=-\frac{15}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{15}i}{2} x=-\frac{\sqrt{15}i}{2}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 72, the least common multiple of 9,24,2.

$$8\left(17x-34x^{2}\right)+72\left(\frac{1}{2}-2x\right)^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use the distributive property to multiply 8 by 17x-34x^{2}.

$$136x-272x^{2}+72\left(\frac{1}{2}-2x\right)^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\frac{1}{2}-2x\right)^{2}.

$$136x-272x^{2}+72\left(\frac{1}{4}-2x+4x^{2}\right)-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use the distributive property to multiply 72 by \frac{1}{4}-2x+4x^{2}.

$$136x-272x^{2}+18-144x+288x^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Combine 136x and -144x to get -8x.

$$-8x-272x^{2}+18+288x^{2}-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Combine -272x^{2} and 288x^{2} to get 16x^{2}.

$$-8x+16x^{2}+18-3=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Subtract 3 from 18 to get 15.

$$-8x+16x^{2}+15=72x^{2}\left(x-\frac{1}{2}\right)-72x\left(x-\frac{1}{3}\right)^{2}$$

Use the distributive property to multiply 72x^{2} by x-\frac{1}{2}.

$$-8x+16x^{2}+15=72x^{3}-36x^{2}-72x\left(x-\frac{1}{3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-\frac{1}{3}\right)^{2}.

$$-8x+16x^{2}+15=72x^{3}-36x^{2}-72x\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right)$$

Add 72x\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right) to both sides.

$$-8x+16x^{2}+15+72x\left(x^{2}-\frac{2}{3}x+\frac{1}{9}\right)=72x^{3}-36x^{2}$$

Use the distributive property to multiply 72x by x^{2}-\frac{2}{3}x+\frac{1}{9}.

$$-8x+16x^{2}+15+72x^{3}-48x^{2}+8x=72x^{3}-36x^{2}$$

Combine 16x^{2} and -48x^{2} to get -32x^{2}.

$$-8x-32x^{2}+15+72x^{3}+8x=72x^{3}-36x^{2}$$

Combine -8x and 8x to get 0.

$$-32x^{2}+15+72x^{3}=72x^{3}-36x^{2}$$

Subtract 72x^{3} from both sides.

$$-32x^{2}+15+72x^{3}-72x^{3}=-36x^{2}$$

Combine 72x^{3} and -72x^{3} to get 0.

$$-32x^{2}+15=-36x^{2}$$

Add 36x^{2} to both sides.

$$-32x^{2}+15+36x^{2}=0$$

Combine -32x^{2} and 36x^{2} to get 4x^{2}.

$$4x^{2}+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 0 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 4\times 15}}{2\times 4}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 4\times 15}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{0±\sqrt{-16\times 15}}{2\times 4}$$

Multiply -16 times 15.

$$x=\frac{0±\sqrt{-240}}{2\times 4}$$

Take the square root of -240.

$$x=\frac{0±4\sqrt{15}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{0±4\sqrt{15}i}{8}$$

Now solve the equation x=\frac{0±4\sqrt{15}i}{8} when ± is plus.

$$x=\frac{\sqrt{15}i}{2}$$

Now solve the equation x=\frac{0±4\sqrt{15}i}{8} when ± is minus.

$$x=-\frac{\sqrt{15}i}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{15}i}{2} x=-\frac{\sqrt{15}i}{2}$$