Solve 2x/1x-x/2+x+1/4=6x

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

Solve for x

$$x = \frac{\sqrt{593} + 25}{16} \approx 3.084474458$$

$$x=\frac{25-\sqrt{593}}{16}\approx 0.040525542$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$4\times 2xx-2x+x+1=24x$$

Multiply 4 and 2 to get 8.

$$8xx-2x+x+1=24x$$

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

$$8x^{2}-2x+x+1=24x$$

Combine -2x and x to get -x.

$$8x^{2}-x+1=24x$$

Subtract 24x from both sides.

$$8x^{2}-x+1-24x=0$$

Combine -x and -24x to get -25x.

$$8x^{2}-25x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -25 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 8}}{2\times 8}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 8}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-25\right)±\sqrt{625-32}}{2\times 8}$$

Add 625 to -32.

$$x=\frac{-\left(-25\right)±\sqrt{593}}{2\times 8}$$

The opposite of -25 is 25.

$$x=\frac{25±\sqrt{593}}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{25±\sqrt{593}}{16}$$

Now solve the equation x=\frac{25±\sqrt{593}}{16} when ± is plus. Add 25 to \sqrt{593}.

$$x=\frac{\sqrt{593}+25}{16}$$

Now solve the equation x=\frac{25±\sqrt{593}}{16} when ± is minus. Subtract \sqrt{593} from 25.

$$x=\frac{25-\sqrt{593}}{16}$$

The equation is now solved.

$$x=\frac{\sqrt{593}+25}{16} x=\frac{25-\sqrt{593}}{16}$$

Steps for Completing the Square

Multiply both sides of the equation by 4, the least common multiple of 2,4.

$$4\times 2xx-2x+x+1=24x$$

Multiply 4 and 2 to get 8.

$$8xx-2x+x+1=24x$$

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

$$8x^{2}-2x+x+1=24x$$

Combine -2x and x to get -x.

$$8x^{2}-x+1=24x$$

Subtract 24x from both sides.

$$8x^{2}-x+1-24x=0$$

Combine -x and -24x to get -25x.

$$8x^{2}-25x+1=0$$

Subtract 1 from both sides. Anything subtracted from zero gives its negation.

$$8x^{2}-25x=-1$$

Divide both sides by 8.

$$\frac{8x^{2}-25x}{8}=-\frac{1}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}-\frac{25}{8}x=-\frac{1}{8}$$

Divide -\frac{25}{8}, the coefficient of the x term, by 2 to get -\frac{25}{16}. Then add the square of -\frac{25}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{25}{8}x+\left(-\frac{25}{16}\right)^{2}=-\frac{1}{8}+\left(-\frac{25}{16}\right)^{2}$$

Square -\frac{25}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{25}{8}x+\frac{625}{256}=-\frac{1}{8}+\frac{625}{256}$$

Add -\frac{1}{8} to \frac{625}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{25}{8}x+\frac{625}{256}=\frac{593}{256}$$

Factor x^{2}-\frac{25}{8}x+\frac{625}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{25}{16}\right)^{2}=\frac{593}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{25}{16}\right)^{2}}=\sqrt{\frac{593}{256}}$$

Simplify.

$$x-\frac{25}{16}=\frac{\sqrt{593}}{16} x-\frac{25}{16}=-\frac{\sqrt{593}}{16}$$

Add \frac{25}{16} to both sides of the equation.

$$x=\frac{\sqrt{593}+25}{16} x=\frac{25-\sqrt{593}}{16}$$