Solve 5/x+6=x-2/x

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

Solve for x

$$x=-1$$

$$x=7$$

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 x.

$$5+x\times 6=xx-2$$

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

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

Subtract x^{2} from both sides.

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

Add 2 to both sides.

$$5+x\times 6-x^{2}+2=0$$

Add 5 and 2 to get 7.

$$7+x\times 6-x^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-x^{2}+6x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 6 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Square 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\times 7}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-6±\sqrt{36+4\times 7}}{2\left(-1\right)}$$

Multiply 4 times 7.

$$x=\frac{-6±\sqrt{36+28}}{2\left(-1\right)}$$

Add 36 to 28.

$$x=\frac{-6±\sqrt{64}}{2\left(-1\right)}$$

Take the square root of 64.

$$x=\frac{-6±8}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-6±8}{-2}$$

Now solve the equation x=\frac{-6±8}{-2} when ± is plus. Add -6 to 8.

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

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{-6±8}{-2} when ± is minus. Subtract 8 from -6.

$$x=-\frac{14}{-2}$$

Divide -14 by -2.

$$x=7$$

The equation is now solved.

$$x=-1 x=7$$

Steps for Completing the Square

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

$$5+x\times 6=xx-2$$

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

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

Subtract x^{2} from both sides.

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

Subtract 5 from both sides.

$$x\times 6-x^{2}=-2-5$$

Subtract 5 from -2 to get -7.

$$x\times 6-x^{2}=-7$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-x^{2}+6x=-7$$

Divide both sides by -1.

$$\frac{-x^{2}+6x}{-1}=-\frac{7}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{6}{-1}x=-\frac{7}{-1}$$

Divide 6 by -1.

$$x^{2}-6x=-\frac{7}{-1}$$

Divide -7 by -1.

$$x^{2}-6x=7$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=7+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=7+9$$

Add 7 to 9.

$$x^{2}-6x+9=16$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{16}$$

Simplify.

$$x-3=4 x-3=-4$$

Add 3 to both sides of the equation.

$$x=7 x=-1$$