Solve 2^152m^2-2m-1=0

$$2 ^ { 1 } 52 m ^ { 2 } - 2 m - 1 = 0$$

Solve for m

$$m=\frac{\sqrt{105}+1}{104}\approx 0.108143757$$

$$m=\frac{1-\sqrt{105}}{104}\approx -0.088912988$$

Steps Using the Quadratic Formula

Calculate 2 to the power of 1 and get 2.

$$2\times 52m^{2}-2m-1=0$$

Multiply 2 and 52 to get 104.

$$104m^{2}-2m-1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 104 for a, -2 for b, and -1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$m=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 104\left(-1\right)}}{2\times 104}$$

Square -2.

$$m=\frac{-\left(-2\right)±\sqrt{4-4\times 104\left(-1\right)}}{2\times 104}$$

Multiply -4 times 104.

$$m=\frac{-\left(-2\right)±\sqrt{4-416\left(-1\right)}}{2\times 104}$$

Multiply -416 times -1.

$$m=\frac{-\left(-2\right)±\sqrt{4+416}}{2\times 104}$$

Add 4 to 416.

$$m=\frac{-\left(-2\right)±\sqrt{420}}{2\times 104}$$

Take the square root of 420.

$$m=\frac{-\left(-2\right)±2\sqrt{105}}{2\times 104}$$

The opposite of -2 is 2.

$$m=\frac{2±2\sqrt{105}}{2\times 104}$$

Multiply 2 times 104.

$$m=\frac{2±2\sqrt{105}}{208}$$

Now solve the equation m=\frac{2±2\sqrt{105}}{208} when ± is plus. Add 2 to 2\sqrt{105}.

$$m=\frac{2\sqrt{105}+2}{208}$$

Divide 2+2\sqrt{105} by 208.

$$m=\frac{\sqrt{105}+1}{104}$$

Now solve the equation m=\frac{2±2\sqrt{105}}{208} when ± is minus. Subtract 2\sqrt{105} from 2.

$$m=\frac{2-2\sqrt{105}}{208}$$

Divide 2-2\sqrt{105} by 208.

$$m=\frac{1-\sqrt{105}}{104}$$

The equation is now solved.

$$m=\frac{\sqrt{105}+1}{104} m=\frac{1-\sqrt{105}}{104}$$

Steps for Completing the Square

Calculate 2 to the power of 1 and get 2.

$$2\times 52m^{2}-2m-1=0$$

Multiply 2 and 52 to get 104.

$$104m^{2}-2m-1=0$$

Add 1 to both sides. Anything plus zero gives itself.

$$104m^{2}-2m=1$$

Divide both sides by 104.

$$\frac{104m^{2}-2m}{104}=\frac{1}{104}$$

Dividing by 104 undoes the multiplication by 104.

$$m^{2}+\left(-\frac{2}{104}\right)m=\frac{1}{104}$$

Reduce the fraction \frac{-2}{104} to lowest terms by extracting and canceling out 2.

$$m^{2}-\frac{1}{52}m=\frac{1}{104}$$

Divide -\frac{1}{52}, the coefficient of the x term, by 2 to get -\frac{1}{104}. Then add the square of -\frac{1}{104} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$m^{2}-\frac{1}{52}m+\left(-\frac{1}{104}\right)^{2}=\frac{1}{104}+\left(-\frac{1}{104}\right)^{2}$$

Square -\frac{1}{104} by squaring both the numerator and the denominator of the fraction.

$$m^{2}-\frac{1}{52}m+\frac{1}{10816}=\frac{1}{104}+\frac{1}{10816}$$

Add \frac{1}{104} to \frac{1}{10816} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$m^{2}-\frac{1}{52}m+\frac{1}{10816}=\frac{105}{10816}$$

Factor m^{2}-\frac{1}{52}m+\frac{1}{10816}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(m-\frac{1}{104}\right)^{2}=\frac{105}{10816}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(m-\frac{1}{104}\right)^{2}}=\sqrt{\frac{105}{10816}}$$

Simplify.

$$m-\frac{1}{104}=\frac{\sqrt{105}}{104} m-\frac{1}{104}=-\frac{\sqrt{105}}{104}$$

Add \frac{1}{104} to both sides of the equation.

$$m=\frac{\sqrt{105}+1}{104} m=\frac{1-\sqrt{105}}{104}$$