Solve 5/x-6/x^3-2x^2=x^2+5x-6/x^3-2{x^2}

$$\frac{ 5 }{ x } - \frac{ 6 }{ { x }^{ 3 } -2 { x }^{ 2 } } = \frac{ { x }^{ 2 } +5x-6 }{ { x }^{ 3 } -2 { x }^{ 2 } }$$

Solve for x

$$x = \frac{15}{4} = 3\frac{3}{4} = 3.75$$

Steps Using Factoring

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

$$\left(x-2\right)x\times 5-6=x^{2}+5x-6$$

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

$$\left(x^{2}-2x\right)\times 5-6=x^{2}+5x-6$$

Use the distributive property to multiply x^{2}-2x by 5.

$$5x^{2}-10x-6=x^{2}+5x-6$$

Subtract x^{2} from both sides.

$$5x^{2}-10x-6-x^{2}=5x-6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-10x-6=5x-6$$

Subtract 5x from both sides.

$$4x^{2}-10x-6-5x=-6$$

Combine -10x and -5x to get -15x.

$$4x^{2}-15x-6=-6$$

Add 6 to both sides.

$$4x^{2}-15x-6+6=0$$

Add -6 and 6 to get 0.

$$4x^{2}-15x=0$$

Factor out x.

$$x\left(4x-15\right)=0$$

To find equation solutions, solve x=0 and 4x-15=0.

$$x=0 x=\frac{15}{4}$$

Variable x cannot be equal to 0.

$$x=\frac{15}{4}$$

Steps Using the Quadratic Formula

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

$$\left(x-2\right)x\times 5-6=x^{2}+5x-6$$

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

$$\left(x^{2}-2x\right)\times 5-6=x^{2}+5x-6$$

Use the distributive property to multiply x^{2}-2x by 5.

$$5x^{2}-10x-6=x^{2}+5x-6$$

Subtract x^{2} from both sides.

$$5x^{2}-10x-6-x^{2}=5x-6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-10x-6=5x-6$$

Subtract 5x from both sides.

$$4x^{2}-10x-6-5x=-6$$

Combine -10x and -5x to get -15x.

$$4x^{2}-15x-6=-6$$

Add 6 to both sides.

$$4x^{2}-15x-6+6=0$$

Add -6 and 6 to get 0.

$$4x^{2}-15x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -15 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}}}{2\times 4}$$

Take the square root of \left(-15\right)^{2}.

$$x=\frac{-\left(-15\right)±15}{2\times 4}$$

The opposite of -15 is 15.

$$x=\frac{15±15}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{15±15}{8}$$

Now solve the equation x=\frac{15±15}{8} when ± is plus. Add 15 to 15.

$$x=\frac{30}{8}$$

Reduce the fraction \frac{30}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{15}{4}$$

Now solve the equation x=\frac{15±15}{8} when ± is minus. Subtract 15 from 15.

$$x=\frac{0}{8}$$

Divide 0 by 8.

$$x=0$$

The equation is now solved.

$$x=\frac{15}{4} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{15}{4}$$

Steps for Completing the Square

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

$$\left(x-2\right)x\times 5-6=x^{2}+5x-6$$

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

$$\left(x^{2}-2x\right)\times 5-6=x^{2}+5x-6$$

Use the distributive property to multiply x^{2}-2x by 5.

$$5x^{2}-10x-6=x^{2}+5x-6$$

Subtract x^{2} from both sides.

$$5x^{2}-10x-6-x^{2}=5x-6$$

Combine 5x^{2} and -x^{2} to get 4x^{2}.

$$4x^{2}-10x-6=5x-6$$

Subtract 5x from both sides.

$$4x^{2}-10x-6-5x=-6$$

Combine -10x and -5x to get -15x.

$$4x^{2}-15x-6=-6$$

Add 6 to both sides.

$$4x^{2}-15x=-6+6$$

Add -6 and 6 to get 0.

$$4x^{2}-15x=0$$

Divide both sides by 4.

$$\frac{4x^{2}-15x}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{15}{4}x=\frac{0}{4}$$

Divide 0 by 4.

$$x^{2}-\frac{15}{4}x=0$$

Divide -\frac{15}{4}, the coefficient of the x term, by 2 to get -\frac{15}{8}. Then add the square of -\frac{15}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{4}x+\left(-\frac{15}{8}\right)^{2}=\left(-\frac{15}{8}\right)^{2}$$

Square -\frac{15}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{4}x+\frac{225}{64}=\frac{225}{64}$$

Factor x^{2}-\frac{15}{4}x+\frac{225}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{8}\right)^{2}=\frac{225}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{8}\right)^{2}}=\sqrt{\frac{225}{64}}$$

Simplify.

$$x-\frac{15}{8}=\frac{15}{8} x-\frac{15}{8}=-\frac{15}{8}$$

Add \frac{15}{8} to both sides of the equation.

$$x=\frac{15}{4} x=0$$

Variable x cannot be equal to 0.

$$x=\frac{15}{4}$$