Solve left(x-2right)^3-x^3=x^2-4

$${ \left(x-2 \right) }^{ 3 } - { x }^{ 3 } = { x }^{ 2 } -4$$

Solve for x

$$x = \frac{2 \sqrt{2} + 6}{7} \approx 1.261203875$$

$$x=\frac{6-2\sqrt{2}}{7}\approx 0.453081839$$

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}.

$$x^{3}-6x^{2}+12x-8-x^{3}=x^{2}-4$$

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

$$-6x^{2}+12x-8=x^{2}-4$$

Subtract x^{2} from both sides.

$$-6x^{2}+12x-8-x^{2}=-4$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$-7x^{2}+12x-8=-4$$

Add 4 to both sides.

$$-7x^{2}+12x-8+4=0$$

Add -8 and 4 to get -4.

$$-7x^{2}+12x-4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -7 for a, 12 for b, and -4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-7\right)\left(-4\right)}}{2\left(-7\right)}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\left(-7\right)\left(-4\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{-12±\sqrt{144+28\left(-4\right)}}{2\left(-7\right)}$$

Multiply 28 times -4.

$$x=\frac{-12±\sqrt{144-112}}{2\left(-7\right)}$$

Add 144 to -112.

$$x=\frac{-12±\sqrt{32}}{2\left(-7\right)}$$

Take the square root of 32.

$$x=\frac{-12±4\sqrt{2}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{-12±4\sqrt{2}}{-14}$$

Now solve the equation x=\frac{-12±4\sqrt{2}}{-14} when ± is plus. Add -12 to 4\sqrt{2}.

$$x=\frac{4\sqrt{2}-12}{-14}$$

Divide -12+4\sqrt{2} by -14.

$$x=\frac{6-2\sqrt{2}}{7}$$

Now solve the equation x=\frac{-12±4\sqrt{2}}{-14} when ± is minus. Subtract 4\sqrt{2} from -12.

$$x=\frac{-4\sqrt{2}-12}{-14}$$

Divide -12-4\sqrt{2} by -14.

$$x=\frac{2\sqrt{2}+6}{7}$$

The equation is now solved.

$$x=\frac{6-2\sqrt{2}}{7} x=\frac{2\sqrt{2}+6}{7}$$

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}.

$$x^{3}-6x^{2}+12x-8-x^{3}=x^{2}-4$$

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

$$-6x^{2}+12x-8=x^{2}-4$$

Subtract x^{2} from both sides.

$$-6x^{2}+12x-8-x^{2}=-4$$

Combine -6x^{2} and -x^{2} to get -7x^{2}.

$$-7x^{2}+12x-8=-4$$

Add 8 to both sides.

$$-7x^{2}+12x=-4+8$$

Add -4 and 8 to get 4.

$$-7x^{2}+12x=4$$

Divide both sides by -7.

$$\frac{-7x^{2}+12x}{-7}=\frac{4}{-7}$$

Dividing by -7 undoes the multiplication by -7.

$$x^{2}+\frac{12}{-7}x=\frac{4}{-7}$$

Divide 12 by -7.

$$x^{2}-\frac{12}{7}x=\frac{4}{-7}$$

Divide 4 by -7.

$$x^{2}-\frac{12}{7}x=-\frac{4}{7}$$

Divide -\frac{12}{7}, the coefficient of the x term, by 2 to get -\frac{6}{7}. Then add the square of -\frac{6}{7} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{12}{7}x+\left(-\frac{6}{7}\right)^{2}=-\frac{4}{7}+\left(-\frac{6}{7}\right)^{2}$$

Square -\frac{6}{7} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=-\frac{4}{7}+\frac{36}{49}$$

Add -\frac{4}{7} to \frac{36}{49} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{12}{7}x+\frac{36}{49}=\frac{8}{49}$$

Factor x^{2}-\frac{12}{7}x+\frac{36}{49}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{6}{7}\right)^{2}=\frac{8}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{6}{7}\right)^{2}}=\sqrt{\frac{8}{49}}$$

Simplify.

$$x-\frac{6}{7}=\frac{2\sqrt{2}}{7} x-\frac{6}{7}=-\frac{2\sqrt{2}}{7}$$

Add \frac{6}{7} to both sides of the equation.

$$x=\frac{2\sqrt{2}+6}{7} x=\frac{6-2\sqrt{2}}{7}$$