Solve x^3-left(x-2right)^3=256

$${ x }^{ 3 } - { \left(x-2 \right) }^{ 3 } =256$$

Solve for x

$$x=\frac{\sqrt{381}}{3}+1\approx 7.506407099$$

$$x=-\frac{\sqrt{381}}{3}+1\approx -5.506407099$$

Steps Using the Quadratic Formula

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

$$x^{3}-\left(x^{3}-6x^{2}+12x-8\right)=256$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$x^{3}-x^{3}+6x^{2}-12x+8=256$$

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

$$6x^{2}-12x+8=256$$

Subtract 256 from both sides.

$$6x^{2}-12x+8-256=0$$

Subtract 256 from 8 to get -248.

$$6x^{2}-12x-248=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -12 for b, and -248 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\left(-248\right)}}{2\times 6}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 6\left(-248\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-12\right)±\sqrt{144-24\left(-248\right)}}{2\times 6}$$

Multiply -24 times -248.

$$x=\frac{-\left(-12\right)±\sqrt{144+5952}}{2\times 6}$$

Add 144 to 5952.

$$x=\frac{-\left(-12\right)±\sqrt{6096}}{2\times 6}$$

Take the square root of 6096.

$$x=\frac{-\left(-12\right)±4\sqrt{381}}{2\times 6}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{381}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{12±4\sqrt{381}}{12}$$

Now solve the equation x=\frac{12±4\sqrt{381}}{12} when ± is plus. Add 12 to 4\sqrt{381}.

$$x=\frac{4\sqrt{381}+12}{12}$$

Divide 12+4\sqrt{381} by 12.

$$x=\frac{\sqrt{381}}{3}+1$$

Now solve the equation x=\frac{12±4\sqrt{381}}{12} when ± is minus. Subtract 4\sqrt{381} from 12.

$$x=\frac{12-4\sqrt{381}}{12}$$

Divide 12-4\sqrt{381} by 12.

$$x=-\frac{\sqrt{381}}{3}+1$$

The equation is now solved.

$$x=\frac{\sqrt{381}}{3}+1 x=-\frac{\sqrt{381}}{3}+1$$

Steps for Completing the Square

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

$$x^{3}-\left(x^{3}-6x^{2}+12x-8\right)=256$$

To find the opposite of x^{3}-6x^{2}+12x-8, find the opposite of each term.

$$x^{3}-x^{3}+6x^{2}-12x+8=256$$

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

$$6x^{2}-12x+8=256$$

Subtract 8 from both sides.

$$6x^{2}-12x=256-8$$

Subtract 8 from 256 to get 248.

$$6x^{2}-12x=248$$

Divide both sides by 6.

$$\frac{6x^{2}-12x}{6}=\frac{248}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{12}{6}\right)x=\frac{248}{6}$$

Divide -12 by 6.

$$x^{2}-2x=\frac{248}{6}$$

Reduce the fraction \frac{248}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}-2x=\frac{124}{3}$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=\frac{124}{3}+1$$

Add \frac{124}{3} to 1.

$$x^{2}-2x+1=\frac{127}{3}$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=\frac{127}{3}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{127}{3}}$$

Simplify.

$$x-1=\frac{\sqrt{381}}{3} x-1=-\frac{\sqrt{381}}{3}$$

Add 1 to both sides of the equation.

$$x=\frac{\sqrt{381}}{3}+1 x=-\frac{\sqrt{381}}{3}+1$$