Solve 6/x=4x+1

$$\frac{ 6 }{ x } =4x+1$$

Solve for x

$$x = \frac{\sqrt{97} - 1}{8} \approx 1.106107225$$

$$x=\frac{-\sqrt{97}-1}{8}\approx -1.356107225$$

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.

$$6=4xx+x$$

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

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 6 from both sides.

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

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

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

Square 1.

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

Multiply -4 times 4.

$$x=\frac{-1±\sqrt{1-16\left(-6\right)}}{2\times 4}$$

Multiply -16 times -6.

$$x=\frac{-1±\sqrt{1+96}}{2\times 4}$$

Add 1 to 96.

$$x=\frac{-1±\sqrt{97}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-1±\sqrt{97}}{8}$$

Now solve the equation x=\frac{-1±\sqrt{97}}{8} when ± is plus. Add -1 to \sqrt{97}.

$$x=\frac{\sqrt{97}-1}{8}$$

Now solve the equation x=\frac{-1±\sqrt{97}}{8} when ± is minus. Subtract \sqrt{97} from -1.

$$x=\frac{-\sqrt{97}-1}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{97}-1}{8} x=\frac{-\sqrt{97}-1}{8}$$

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.

$$6=4xx+x$$

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

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

Swap sides so that all variable terms are on the left hand side.

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

Divide both sides by 4.

$$\frac{4x^{2}+x}{4}=\frac{6}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{1}{4}x=\frac{6}{4}$$

Reduce the fraction \frac{6}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{4}x=\frac{3}{2}$$

Divide \frac{1}{4}, the coefficient of the x term, by 2 to get \frac{1}{8}. Then add the square of \frac{1}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{4}x+\left(\frac{1}{8}\right)^{2}=\frac{3}{2}+\left(\frac{1}{8}\right)^{2}$$

Square \frac{1}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{3}{2}+\frac{1}{64}$$

Add \frac{3}{2} to \frac{1}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{1}{4}x+\frac{1}{64}=\frac{97}{64}$$

Factor x^{2}+\frac{1}{4}x+\frac{1}{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}.

$$\left(x+\frac{1}{8}\right)^{2}=\frac{97}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{8}\right)^{2}}=\sqrt{\frac{97}{64}}$$

Simplify.

$$x+\frac{1}{8}=\frac{\sqrt{97}}{8} x+\frac{1}{8}=-\frac{\sqrt{97}}{8}$$

Subtract \frac{1}{8} from both sides of the equation.

$$x=\frac{\sqrt{97}-1}{8} x=\frac{-\sqrt{97}-1}{8}$$