Solve (x-2)(x^2+2x+4)-x^2(x-18)=0

$$( x - 2 ) ( x ^ { 2 } + 2 x + 4 ) - x ^ { 2 } ( x - 18 ) = 0$$

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=\frac{2}{3}\approx 0.666666667$$

Steps Using Difference of Squares

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

$$x^{3}-8-x^{2}\left(x-18\right)=0$$

Use the distributive property to multiply x^{2} by x-18.

$$x^{3}-8-\left(x^{3}-18x^{2}\right)=0$$

To find the opposite of x^{3}-18x^{2}, find the opposite of each term.

$$x^{3}-8-x^{3}+18x^{2}=0$$

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

$$-8+18x^{2}=0$$

Divide both sides by 2.

$$-4+9x^{2}=0$$

Consider -4+9x^{2}. Rewrite -4+9x^{2} as \left(3x\right)^{2}-2^{2}. The difference of squares can be factored using the rule: a^{2}-b^{2}=\left(a-b\right)\left(a+b\right).

$$\left(3x-2\right)\left(3x+2\right)=0$$

To find equation solutions, solve 3x-2=0 and 3x+2=0.

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

Steps by Finding Square Root

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

$$x^{3}-8-x^{2}\left(x-18\right)=0$$

Use the distributive property to multiply x^{2} by x-18.

$$x^{3}-8-\left(x^{3}-18x^{2}\right)=0$$

To find the opposite of x^{3}-18x^{2}, find the opposite of each term.

$$x^{3}-8-x^{3}+18x^{2}=0$$

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

$$-8+18x^{2}=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$18x^{2}=8$$

Divide both sides by 18.

$$x^{2}=\frac{8}{18}$$

Reduce the fraction \frac{8}{18} to lowest terms by extracting and canceling out 2.

$$x^{2}=\frac{4}{9}$$

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

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

$$x^{3}-8-x^{2}\left(x-18\right)=0$$

Use the distributive property to multiply x^{2} by x-18.

$$x^{3}-8-\left(x^{3}-18x^{2}\right)=0$$

To find the opposite of x^{3}-18x^{2}, find the opposite of each term.

$$x^{3}-8-x^{3}+18x^{2}=0$$

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

$$-8+18x^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$18x^{2}-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 18 for a, 0 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 18\left(-8\right)}}{2\times 18}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 18\left(-8\right)}}{2\times 18}$$

Multiply -4 times 18.

$$x=\frac{0±\sqrt{-72\left(-8\right)}}{2\times 18}$$

Multiply -72 times -8.

$$x=\frac{0±\sqrt{576}}{2\times 18}$$

Take the square root of 576.

$$x=\frac{0±24}{2\times 18}$$

Multiply 2 times 18.

$$x=\frac{0±24}{36}$$

Now solve the equation x=\frac{0±24}{36} when ± is plus. Reduce the fraction \frac{24}{36} to lowest terms by extracting and canceling out 12.

$$x=\frac{2}{3}$$

Now solve the equation x=\frac{0±24}{36} when ± is minus. Reduce the fraction \frac{-24}{36} to lowest terms by extracting and canceling out 12.

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

The equation is now solved.

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