Solve 7x+1=6/x

$$7 x + 1 = \frac { 6 } { x }$$

Solve for x

$$x=-1$$

$$x=\frac{6}{7}\approx 0.857142857$$

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.

$$7xx+x=6$$

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

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

Subtract 6 from both sides.

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

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

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

Square 1.

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

Multiply -4 times 7.

$$x=\frac{-1±\sqrt{1-28\left(-6\right)}}{2\times 7}$$

Multiply -28 times -6.

$$x=\frac{-1±\sqrt{1+168}}{2\times 7}$$

Add 1 to 168.

$$x=\frac{-1±\sqrt{169}}{2\times 7}$$

Take the square root of 169.

$$x=\frac{-1±13}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{-1±13}{14}$$

Now solve the equation x=\frac{-1±13}{14} when ± is plus. Add -1 to 13.

$$x=\frac{12}{14}$$

Reduce the fraction \frac{12}{14} to lowest terms by extracting and canceling out 2.

$$x=\frac{6}{7}$$

Now solve the equation x=\frac{-1±13}{14} when ± is minus. Subtract 13 from -1.

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

Divide -14 by 14.

$$x=-1$$

The equation is now solved.

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

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.

$$7xx+x=6$$

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

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

Divide both sides by 7.

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

Dividing by 7 undoes the multiplication by 7.

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

Divide \frac{1}{7}, the coefficient of the x term, by 2 to get \frac{1}{14}. Then add the square of \frac{1}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{7}x+\left(\frac{1}{14}\right)^{2}=\frac{6}{7}+\left(\frac{1}{14}\right)^{2}$$

Square \frac{1}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{6}{7}+\frac{1}{196}$$

Add \frac{6}{7} to \frac{1}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{169}{196}$$

Factor x^{2}+\frac{1}{7}x+\frac{1}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{14}\right)^{2}=\frac{169}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{14}\right)^{2}}=\sqrt{\frac{169}{196}}$$

Simplify.

$$x+\frac{1}{14}=\frac{13}{14} x+\frac{1}{14}=-\frac{13}{14}$$

Subtract \frac{1}{14} from both sides of the equation.

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