Solve x^-1+9x/x=43/4

$$\frac { x ^ { - 1 } + 9 x } { x } = \frac { 43 } { 4 }$$

Solve for x

$$x=\frac{2\sqrt{7}}{7}\approx 0.755928946$$

$$x=-\frac{2\sqrt{7}}{7}\approx -0.755928946$$

Steps by Finding Square Root

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x, the least common multiple of x,4.

$$4\left(x^{-1}+9x\right)=43x$$

Use the distributive property to multiply 4 by x^{-1}+9x.

$$4x^{-1}+36x=43x$$

Subtract 43x from both sides.

$$4x^{-1}+36x-43x=0$$

Combine 36x and -43x to get -7x.

$$4x^{-1}-7x=0$$

Reorder the terms.

$$-7x+4\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-7xx+4\times 1=0$$

Multiply x and x to get x^{2}.

$$-7x^{2}+4\times 1=0$$

Multiply 4 and 1 to get 4.

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

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by -7.

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

Fraction \frac{-4}{-7} can be simplified to \frac{4}{7} by removing the negative sign from both the numerator and the denominator.

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

Take the square root of both sides of the equation.

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

Steps Using the Quadratic Formula

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4x, the least common multiple of x,4.

$$4\left(x^{-1}+9x\right)=43x$$

Use the distributive property to multiply 4 by x^{-1}+9x.

$$4x^{-1}+36x=43x$$

Subtract 43x from both sides.

$$4x^{-1}+36x-43x=0$$

Combine 36x and -43x to get -7x.

$$4x^{-1}-7x=0$$

Reorder the terms.

$$-7x+4\times \frac{1}{x}=0$$

Variable x cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by x.

$$-7xx+4\times 1=0$$

Multiply x and x to get x^{2}.

$$-7x^{2}+4\times 1=0$$

Multiply 4 and 1 to get 4.

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

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

$$x=\frac{0±\sqrt{0^{2}-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-7\right)\times 4}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$x=\frac{0±\sqrt{28\times 4}}{2\left(-7\right)}$$

Multiply 28 times 4.

$$x=\frac{0±\sqrt{112}}{2\left(-7\right)}$$

Take the square root of 112.

$$x=\frac{0±4\sqrt{7}}{2\left(-7\right)}$$

Multiply 2 times -7.

$$x=\frac{0±4\sqrt{7}}{-14}$$

Now solve the equation x=\frac{0±4\sqrt{7}}{-14} when ± is plus.

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

Now solve the equation x=\frac{0±4\sqrt{7}}{-14} when ± is minus.

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

The equation is now solved.

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