Solve 2x^2/x^2-3x-3x-1/3x+9=2x^2+1/2x^2-18

$$\frac { 2 x ^ { 2 } } { x ^ { 2 } - 3 x } - \frac { 3 x - 1 } { 3 x + 9 } = \frac { 2 x ^ { 2 } + 1 } { 2 x ^ { 2 } - 18 }$$

Solve for x

$$x=\frac{9}{56}\approx 0.160714286$$

Steps Using Factoring

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-3x,3x+9,2x^{2}-18.

$$\left(6x+18\right)\times 2x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 6x+18 by 2.

$$\left(12x+36\right)x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 12x+36 by x^{2}.

$$12x^{3}+36x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 2x by x-3.

$$12x^{3}+36x^{2}-\left(2x^{2}-6x\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 2x^{2}-6x by 3x-1 and combine like terms.

$$12x^{3}+36x^{2}-\left(6x^{3}-20x^{2}+6x\right)=3x\left(2x^{2}+1\right)$$

To find the opposite of 6x^{3}-20x^{2}+6x, find the opposite of each term.

$$12x^{3}+36x^{2}-6x^{3}+20x^{2}-6x=3x\left(2x^{2}+1\right)$$

Combine 12x^{3} and -6x^{3} to get 6x^{3}.

$$6x^{3}+36x^{2}+20x^{2}-6x=3x\left(2x^{2}+1\right)$$

Combine 36x^{2} and 20x^{2} to get 56x^{2}.

$$6x^{3}+56x^{2}-6x=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 3x by 2x^{2}+1.

$$6x^{3}+56x^{2}-6x=6x^{3}+3x$$

Subtract 6x^{3} from both sides.

$$6x^{3}+56x^{2}-6x-6x^{3}=3x$$

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

$$56x^{2}-6x=3x$$

Subtract 3x from both sides.

$$56x^{2}-6x-3x=0$$

Combine -6x and -3x to get -9x.

$$56x^{2}-9x=0$$

Factor out x.

$$x\left(56x-9\right)=0$$

To find equation solutions, solve x=0 and 56x-9=0.

$$x=0 x=\frac{9}{56}$$

Variable x cannot be equal to 0.

$$x=\frac{9}{56}$$

Steps Using the Quadratic Formula

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-3x,3x+9,2x^{2}-18.

$$\left(6x+18\right)\times 2x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 6x+18 by 2.

$$\left(12x+36\right)x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 12x+36 by x^{2}.

$$12x^{3}+36x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 2x by x-3.

$$12x^{3}+36x^{2}-\left(2x^{2}-6x\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 2x^{2}-6x by 3x-1 and combine like terms.

$$12x^{3}+36x^{2}-\left(6x^{3}-20x^{2}+6x\right)=3x\left(2x^{2}+1\right)$$

To find the opposite of 6x^{3}-20x^{2}+6x, find the opposite of each term.

$$12x^{3}+36x^{2}-6x^{3}+20x^{2}-6x=3x\left(2x^{2}+1\right)$$

Combine 12x^{3} and -6x^{3} to get 6x^{3}.

$$6x^{3}+36x^{2}+20x^{2}-6x=3x\left(2x^{2}+1\right)$$

Combine 36x^{2} and 20x^{2} to get 56x^{2}.

$$6x^{3}+56x^{2}-6x=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 3x by 2x^{2}+1.

$$6x^{3}+56x^{2}-6x=6x^{3}+3x$$

Subtract 6x^{3} from both sides.

$$6x^{3}+56x^{2}-6x-6x^{3}=3x$$

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

$$56x^{2}-6x=3x$$

Subtract 3x from both sides.

$$56x^{2}-6x-3x=0$$

Combine -6x and -3x to get -9x.

$$56x^{2}-9x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 56 for a, -9 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}}}{2\times 56}$$

Take the square root of \left(-9\right)^{2}.

$$x=\frac{-\left(-9\right)±9}{2\times 56}$$

The opposite of -9 is 9.

$$x=\frac{9±9}{2\times 56}$$

Multiply 2 times 56.

$$x=\frac{9±9}{112}$$

Now solve the equation x=\frac{9±9}{112} when ± is plus. Add 9 to 9.

$$x=\frac{18}{112}$$

Reduce the fraction \frac{18}{112} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{56}$$

Now solve the equation x=\frac{9±9}{112} when ± is minus. Subtract 9 from 9.

$$x=\frac{0}{112}$$

Divide 0 by 112.

$$x=0$$

The equation is now solved.

$$x=\frac{9}{56} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{9}{56}$$

Steps for Completing the Square

Variable x cannot be equal to any of the values -3,0,3 since division by zero is not defined. Multiply both sides of the equation by 6x\left(x-3\right)\left(x+3\right), the least common multiple of x^{2}-3x,3x+9,2x^{2}-18.

$$\left(6x+18\right)\times 2x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 6x+18 by 2.

$$\left(12x+36\right)x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 12x+36 by x^{2}.

$$12x^{3}+36x^{2}-2x\left(x-3\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 2x by x-3.

$$12x^{3}+36x^{2}-\left(2x^{2}-6x\right)\left(3x-1\right)=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 2x^{2}-6x by 3x-1 and combine like terms.

$$12x^{3}+36x^{2}-\left(6x^{3}-20x^{2}+6x\right)=3x\left(2x^{2}+1\right)$$

To find the opposite of 6x^{3}-20x^{2}+6x, find the opposite of each term.

$$12x^{3}+36x^{2}-6x^{3}+20x^{2}-6x=3x\left(2x^{2}+1\right)$$

Combine 12x^{3} and -6x^{3} to get 6x^{3}.

$$6x^{3}+36x^{2}+20x^{2}-6x=3x\left(2x^{2}+1\right)$$

Combine 36x^{2} and 20x^{2} to get 56x^{2}.

$$6x^{3}+56x^{2}-6x=3x\left(2x^{2}+1\right)$$

Use the distributive property to multiply 3x by 2x^{2}+1.

$$6x^{3}+56x^{2}-6x=6x^{3}+3x$$

Subtract 6x^{3} from both sides.

$$6x^{3}+56x^{2}-6x-6x^{3}=3x$$

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

$$56x^{2}-6x=3x$$

Subtract 3x from both sides.

$$56x^{2}-6x-3x=0$$

Combine -6x and -3x to get -9x.

$$56x^{2}-9x=0$$

Divide both sides by 56.

$$\frac{56x^{2}-9x}{56}=\frac{0}{56}$$

Dividing by 56 undoes the multiplication by 56.

$$x^{2}-\frac{9}{56}x=\frac{0}{56}$$

Divide 0 by 56.

$$x^{2}-\frac{9}{56}x=0$$

Divide -\frac{9}{56}, the coefficient of the x term, by 2 to get -\frac{9}{112}. Then add the square of -\frac{9}{112} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{56}x+\left(-\frac{9}{112}\right)^{2}=\left(-\frac{9}{112}\right)^{2}$$

Square -\frac{9}{112} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{56}x+\frac{81}{12544}=\frac{81}{12544}$$

Factor x^{2}-\frac{9}{56}x+\frac{81}{12544}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{112}\right)^{2}=\frac{81}{12544}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{112}\right)^{2}}=\sqrt{\frac{81}{12544}}$$

Simplify.

$$x-\frac{9}{112}=\frac{9}{112} x-\frac{9}{112}=-\frac{9}{112}$$

Add \frac{9}{112} to both sides of the equation.

$$x=\frac{9}{56} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{9}{56}$$