Solve 1/x+x=13/4

$$\frac { 1 } { x } + x = \frac { 13 } { 4 }$$

Solve for x

$$x = \frac{\sqrt{105} + 13}{8} \approx 2.905868846$$

$$x=\frac{13-\sqrt{105}}{8}\approx 0.344131154$$

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.

$$4+4xx=13x$$

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

$$4+4x^{2}=13x$$

Subtract 13x from both sides.

$$4+4x^{2}-13x=0$$

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.

$$4x^{2}-13x+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -13 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 4\times 4}}{2\times 4}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 4\times 4}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-13\right)±\sqrt{169-16\times 4}}{2\times 4}$$

Multiply -16 times 4.

$$x=\frac{-\left(-13\right)±\sqrt{169-64}}{2\times 4}$$

Add 169 to -64.

$$x=\frac{-\left(-13\right)±\sqrt{105}}{2\times 4}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{105}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{13±\sqrt{105}}{8}$$

Now solve the equation x=\frac{13±\sqrt{105}}{8} when ± is plus. Add 13 to \sqrt{105}.

$$x=\frac{\sqrt{105}+13}{8}$$

Now solve the equation x=\frac{13±\sqrt{105}}{8} when ± is minus. Subtract \sqrt{105} from 13.

$$x=\frac{13-\sqrt{105}}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{105}+13}{8} x=\frac{13-\sqrt{105}}{8}$$

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 4x, the least common multiple of x,4.

$$4+4xx=13x$$

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

$$4+4x^{2}=13x$$

Subtract 13x from both sides.

$$4+4x^{2}-13x=0$$

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

$$4x^{2}-13x=-4$$

Divide both sides by 4.

$$\frac{4x^{2}-13x}{4}=-\frac{4}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}-\frac{13}{4}x=-\frac{4}{4}$$

Divide -4 by 4.

$$x^{2}-\frac{13}{4}x=-1$$

Divide -\frac{13}{4}, the coefficient of the x term, by 2 to get -\frac{13}{8}. Then add the square of -\frac{13}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{13}{4}x+\left(-\frac{13}{8}\right)^{2}=-1+\left(-\frac{13}{8}\right)^{2}$$

Square -\frac{13}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=-1+\frac{169}{64}$$

Add -1 to \frac{169}{64}.

$$x^{2}-\frac{13}{4}x+\frac{169}{64}=\frac{105}{64}$$

Factor x^{2}-\frac{13}{4}x+\frac{169}{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}.

$$\left(x-\frac{13}{8}\right)^{2}=\frac{105}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{13}{8}\right)^{2}}=\sqrt{\frac{105}{64}}$$

Simplify.

$$x-\frac{13}{8}=\frac{\sqrt{105}}{8} x-\frac{13}{8}=-\frac{\sqrt{105}}{8}$$

Add \frac{13}{8} to both sides of the equation.

$$x=\frac{\sqrt{105}+13}{8} x=\frac{13-\sqrt{105}}{8}$$