Solve 2x^3-12x^2+9x/4x^2(x^2+3)=(x-3)/2{x^2+6x}

$$\frac{ 2 { x }^{ 3 } -12 { x }^{ 2 } +9x }{ 4 { x }^{ 2 } ( { x }^{ 2 } +3) } = \frac{ (x-3) }{ 2 { x }^{ 2 } +6x }$$

Solve for x

$$x = \frac{15}{11} = 1\frac{4}{11} \approx 1.363636364$$

Steps Using Factoring

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+3\right)x^{2}\left(x^{2}+3\right), the least common multiple of 4x^{2}\left(x^{2}+3\right),2x^{2}+6x.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

Use the distributive property to multiply x+3 by 2x^{3}-12x^{2}+9x and combine like terms.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

Use the distributive property to multiply 2x by x^{2}+3.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

Use the distributive property to multiply 2x^{3}+6x by x-3.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Subtract 2x^{4} from both sides.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

Combine 2x^{4} and -2x^{4} to get 0.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

Add 6x^{3} to both sides.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

Combine -6x^{3} and 6x^{3} to get 0.

$$-27x^{2}+27x=6x^{2}-18x$$

Subtract 6x^{2} from both sides.

$$-27x^{2}+27x-6x^{2}=-18x$$

Combine -27x^{2} and -6x^{2} to get -33x^{2}.

$$-33x^{2}+27x=-18x$$

Add 18x to both sides.

$$-33x^{2}+27x+18x=0$$

Combine 27x and 18x to get 45x.

$$-33x^{2}+45x=0$$

Factor out x.

$$x\left(-33x+45\right)=0$$

To find equation solutions, solve x=0 and -33x+45=0.

$$x=0 x=\frac{15}{11}$$

Variable x cannot be equal to 0.

$$x=\frac{15}{11}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+3\right)x^{2}\left(x^{2}+3\right), the least common multiple of 4x^{2}\left(x^{2}+3\right),2x^{2}+6x.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

Use the distributive property to multiply x+3 by 2x^{3}-12x^{2}+9x and combine like terms.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

Use the distributive property to multiply 2x by x^{2}+3.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

Use the distributive property to multiply 2x^{3}+6x by x-3.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Subtract 2x^{4} from both sides.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

Combine 2x^{4} and -2x^{4} to get 0.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

Add 6x^{3} to both sides.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

Combine -6x^{3} and 6x^{3} to get 0.

$$-27x^{2}+27x=6x^{2}-18x$$

Subtract 6x^{2} from both sides.

$$-27x^{2}+27x-6x^{2}=-18x$$

Combine -27x^{2} and -6x^{2} to get -33x^{2}.

$$-33x^{2}+27x=-18x$$

Add 18x to both sides.

$$-33x^{2}+27x+18x=0$$

Combine 27x and 18x to get 45x.

$$-33x^{2}+45x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -33 for a, 45 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-45±\sqrt{45^{2}}}{2\left(-33\right)}$$

Take the square root of 45^{2}.

$$x=\frac{-45±45}{2\left(-33\right)}$$

Multiply 2 times -33.

$$x=\frac{-45±45}{-66}$$

Now solve the equation x=\frac{-45±45}{-66} when ± is plus. Add -45 to 45.

$$x=\frac{0}{-66}$$

Divide 0 by -66.

$$x=0$$

Now solve the equation x=\frac{-45±45}{-66} when ± is minus. Subtract 45 from -45.

$$x=-\frac{90}{-66}$$

Reduce the fraction \frac{-90}{-66} to lowest terms by extracting and canceling out 6.

$$x=\frac{15}{11}$$

The equation is now solved.

$$x=0 x=\frac{15}{11}$$

Variable x cannot be equal to 0.

$$x=\frac{15}{11}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0 since division by zero is not defined. Multiply both sides of the equation by 4\left(x+3\right)x^{2}\left(x^{2}+3\right), the least common multiple of 4x^{2}\left(x^{2}+3\right),2x^{2}+6x.

$$\left(x+3\right)\left(2x^{3}-12x^{2}+9x\right)=2x\left(x^{2}+3\right)\left(x-3\right)$$

Use the distributive property to multiply x+3 by 2x^{3}-12x^{2}+9x and combine like terms.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x\left(x^{2}+3\right)\left(x-3\right)$$

Use the distributive property to multiply 2x by x^{2}+3.

$$2x^{4}-6x^{3}-27x^{2}+27x=\left(2x^{3}+6x\right)\left(x-3\right)$$

Use the distributive property to multiply 2x^{3}+6x by x-3.

$$2x^{4}-6x^{3}-27x^{2}+27x=2x^{4}-6x^{3}+6x^{2}-18x$$

Subtract 2x^{4} from both sides.

$$2x^{4}-6x^{3}-27x^{2}+27x-2x^{4}=-6x^{3}+6x^{2}-18x$$

Combine 2x^{4} and -2x^{4} to get 0.

$$-6x^{3}-27x^{2}+27x=-6x^{3}+6x^{2}-18x$$

Add 6x^{3} to both sides.

$$-6x^{3}-27x^{2}+27x+6x^{3}=6x^{2}-18x$$

Combine -6x^{3} and 6x^{3} to get 0.

$$-27x^{2}+27x=6x^{2}-18x$$

Subtract 6x^{2} from both sides.

$$-27x^{2}+27x-6x^{2}=-18x$$

Combine -27x^{2} and -6x^{2} to get -33x^{2}.

$$-33x^{2}+27x=-18x$$

Add 18x to both sides.

$$-33x^{2}+27x+18x=0$$

Combine 27x and 18x to get 45x.

$$-33x^{2}+45x=0$$

Divide both sides by -33.

$$\frac{-33x^{2}+45x}{-33}=\frac{0}{-33}$$

Dividing by -33 undoes the multiplication by -33.

$$x^{2}+\frac{45}{-33}x=\frac{0}{-33}$$

Reduce the fraction \frac{45}{-33} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{15}{11}x=\frac{0}{-33}$$

Divide 0 by -33.

$$x^{2}-\frac{15}{11}x=0$$

Divide -\frac{15}{11}, the coefficient of the x term, by 2 to get -\frac{15}{22}. Then add the square of -\frac{15}{22} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{11}x+\left(-\frac{15}{22}\right)^{2}=\left(-\frac{15}{22}\right)^{2}$$

Square -\frac{15}{22} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{11}x+\frac{225}{484}=\frac{225}{484}$$

Factor x^{2}-\frac{15}{11}x+\frac{225}{484}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{22}\right)^{2}=\frac{225}{484}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{22}\right)^{2}}=\sqrt{\frac{225}{484}}$$

Simplify.

$$x-\frac{15}{22}=\frac{15}{22} x-\frac{15}{22}=-\frac{15}{22}$$

Add \frac{15}{22} to both sides of the equation.

$$x=\frac{15}{11} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{15}{11}$$