Solve 0=91left(x-39xright)^2-x/26

$$0 = 91 { \left(x-39x \right) }^{ 2 } - \frac{ x }{ 26 }$$

Solve for x

$$x=\frac{1}{3416504}\approx 0.000000293$$

$$x=0$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 26.

$$0=2366\left(x-39x\right)^{2}-x$$

Combine x and -39x to get -38x.

$$0=2366\left(-38x\right)^{2}-x$$

Expand \left(-38x\right)^{2}.

$$0=2366\left(-38\right)^{2}x^{2}-x$$

Calculate -38 to the power of 2 and get 1444.

$$0=2366\times 1444x^{2}-x$$

Multiply 2366 and 1444 to get 3416504.

$$0=3416504x^{2}-x$$

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

$$3416504x^{2}-x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3416504 for a, -1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1}}{2\times 3416504}$$

Take the square root of 1.

$$x=\frac{-\left(-1\right)±1}{2\times 3416504}$$

The opposite of -1 is 1.

$$x=\frac{1±1}{2\times 3416504}$$

Multiply 2 times 3416504.

$$x=\frac{1±1}{6833008}$$

Now solve the equation x=\frac{1±1}{6833008} when ± is plus. Add 1 to 1.

$$x=\frac{2}{6833008}$$

Reduce the fraction \frac{2}{6833008} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{3416504}$$

Now solve the equation x=\frac{1±1}{6833008} when ± is minus. Subtract 1 from 1.

$$x=\frac{0}{6833008}$$

Divide 0 by 6833008.

$$x=0$$

The equation is now solved.

$$x=\frac{1}{3416504} x=0$$

Steps for Completing the Square

Multiply both sides of the equation by 26.

$$0=2366\left(x-39x\right)^{2}-x$$

Combine x and -39x to get -38x.

$$0=2366\left(-38x\right)^{2}-x$$

Expand \left(-38x\right)^{2}.

$$0=2366\left(-38\right)^{2}x^{2}-x$$

Calculate -38 to the power of 2 and get 1444.

$$0=2366\times 1444x^{2}-x$$

Multiply 2366 and 1444 to get 3416504.

$$0=3416504x^{2}-x$$

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

$$3416504x^{2}-x=0$$

Divide both sides by 3416504.

$$\frac{3416504x^{2}-x}{3416504}=\frac{0}{3416504}$$

Dividing by 3416504 undoes the multiplication by 3416504.

$$x^{2}-\frac{1}{3416504}x=\frac{0}{3416504}$$

Divide 0 by 3416504.

$$x^{2}-\frac{1}{3416504}x=0$$

Divide -\frac{1}{3416504}, the coefficient of the x term, by 2 to get -\frac{1}{6833008}. Then add the square of -\frac{1}{6833008} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{1}{3416504}x+\left(-\frac{1}{6833008}\right)^{2}=\left(-\frac{1}{6833008}\right)^{2}$$

Square -\frac{1}{6833008} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{1}{3416504}x+\frac{1}{46689998328064}=\frac{1}{46689998328064}$$

Factor x^{2}-\frac{1}{3416504}x+\frac{1}{46689998328064}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{6833008}\right)^{2}=\frac{1}{46689998328064}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{6833008}\right)^{2}}=\sqrt{\frac{1}{46689998328064}}$$

Simplify.

$$x-\frac{1}{6833008}=\frac{1}{6833008} x-\frac{1}{6833008}=-\frac{1}{6833008}$$

Add \frac{1}{6833008} to both sides of the equation.

$$x=\frac{1}{3416504} x=0$$