Solve (1/2)^-16x/38642136=12x^2

$$\frac { ( 1 / 2 ) ^ { - 16 } x } { 38642136 } = 12 x ^ { 2 }$$

Solve for x

$$x=\frac{2048}{14490801}\approx 0.000141331$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 38642136.

$$\left(\frac{1}{2}\right)^{-16}x=463705632x^{2}$$

Calculate \frac{1}{2} to the power of -16 and get 65536.

$$65536x=463705632x^{2}$$

Subtract 463705632x^{2} from both sides.

$$65536x-463705632x^{2}=0$$

Factor out x.

$$x\left(65536-463705632x\right)=0$$

To find equation solutions, solve x=0 and 65536-463705632x=0.

$$x=0 x=\frac{2048}{14490801}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 38642136.

$$\left(\frac{1}{2}\right)^{-16}x=463705632x^{2}$$

Calculate \frac{1}{2} to the power of -16 and get 65536.

$$65536x=463705632x^{2}$$

Subtract 463705632x^{2} from both sides.

$$65536x-463705632x^{2}=0$$

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.

$$-463705632x^{2}+65536x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -463705632 for a, 65536 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65536±\sqrt{65536^{2}}}{2\left(-463705632\right)}$$

Take the square root of 65536^{2}.

$$x=\frac{-65536±65536}{2\left(-463705632\right)}$$

Multiply 2 times -463705632.

$$x=\frac{-65536±65536}{-927411264}$$

Now solve the equation x=\frac{-65536±65536}{-927411264} when ± is plus. Add -65536 to 65536.

$$x=\frac{0}{-927411264}$$

Divide 0 by -927411264.

$$x=0$$

Now solve the equation x=\frac{-65536±65536}{-927411264} when ± is minus. Subtract 65536 from -65536.

$$x=-\frac{131072}{-927411264}$$

Reduce the fraction \frac{-131072}{-927411264} to lowest terms by extracting and canceling out 64.

$$x=\frac{2048}{14490801}$$

The equation is now solved.

$$x=0 x=\frac{2048}{14490801}$$

Steps for Completing the Square

Multiply both sides of the equation by 38642136.

$$\left(\frac{1}{2}\right)^{-16}x=463705632x^{2}$$

Calculate \frac{1}{2} to the power of -16 and get 65536.

$$65536x=463705632x^{2}$$

Subtract 463705632x^{2} from both sides.

$$65536x-463705632x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-463705632x^{2}+65536x=0$$

Divide both sides by -463705632.

$$\frac{-463705632x^{2}+65536x}{-463705632}=\frac{0}{-463705632}$$

Dividing by -463705632 undoes the multiplication by -463705632.

$$x^{2}+\frac{65536}{-463705632}x=\frac{0}{-463705632}$$

Reduce the fraction \frac{65536}{-463705632} to lowest terms by extracting and canceling out 32.

$$x^{2}-\frac{2048}{14490801}x=\frac{0}{-463705632}$$

Divide 0 by -463705632.

$$x^{2}-\frac{2048}{14490801}x=0$$

Divide -\frac{2048}{14490801}, the coefficient of the x term, by 2 to get -\frac{1024}{14490801}. Then add the square of -\frac{1024}{14490801} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2048}{14490801}x+\left(-\frac{1024}{14490801}\right)^{2}=\left(-\frac{1024}{14490801}\right)^{2}$$

Square -\frac{1024}{14490801} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2048}{14490801}x+\frac{1048576}{209983313621601}=\frac{1048576}{209983313621601}$$

Factor x^{2}-\frac{2048}{14490801}x+\frac{1048576}{209983313621601}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1024}{14490801}\right)^{2}=\frac{1048576}{209983313621601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1024}{14490801}\right)^{2}}=\sqrt{\frac{1048576}{209983313621601}}$$

Simplify.

$$x-\frac{1024}{14490801}=\frac{1024}{14490801} x-\frac{1024}{14490801}=-\frac{1024}{14490801}$$

Add \frac{1024}{14490801} to both sides of the equation.

$$x=\frac{2048}{14490801} x=0$$