Solve 0.0001x^2+x-192=0

$$0.0001 x ^ { 2 } + x - 192 = 0$$

Solve for x

$$x=200\sqrt{673}-5000\approx 188.448708429$$

$$x=-200\sqrt{673}-5000\approx -10188.448708429$$

Steps Using the Quadratic Formula

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.

$$0.0001x^{2}+x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.0001 for a, 1 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 0.0001\left(-192\right)}}{2\times 0.0001}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\times 0.0001\left(-192\right)}}{2\times 0.0001}$$

Multiply -4 times 0.0001.

$$x=\frac{-1±\sqrt{1-0.0004\left(-192\right)}}{2\times 0.0001}$$

Multiply -0.0004 times -192.

$$x=\frac{-1±\sqrt{1+0.0768}}{2\times 0.0001}$$

Add 1 to 0.0768.

$$x=\frac{-1±\sqrt{1.0768}}{2\times 0.0001}$$

Take the square root of 1.0768.

$$x=\frac{-1±\frac{\sqrt{673}}{25}}{2\times 0.0001}$$

Multiply 2 times 0.0001.

$$x=\frac{-1±\frac{\sqrt{673}}{25}}{0.0002}$$

Now solve the equation x=\frac{-1±\frac{\sqrt{673}}{25}}{0.0002} when ± is plus. Add -1 to \frac{\sqrt{673}}{25}.

$$x=\frac{\frac{\sqrt{673}}{25}-1}{0.0002}$$

Divide -1+\frac{\sqrt{673}}{25} by 0.0002 by multiplying -1+\frac{\sqrt{673}}{25} by the reciprocal of 0.0002.

$$x=200\sqrt{673}-5000$$

Now solve the equation x=\frac{-1±\frac{\sqrt{673}}{25}}{0.0002} when ± is minus. Subtract \frac{\sqrt{673}}{25} from -1.

$$x=\frac{-\frac{\sqrt{673}}{25}-1}{0.0002}$$

Divide -1-\frac{\sqrt{673}}{25} by 0.0002 by multiplying -1-\frac{\sqrt{673}}{25} by the reciprocal of 0.0002.

$$x=-200\sqrt{673}-5000$$

The equation is now solved.

$$x=200\sqrt{673}-5000 x=-200\sqrt{673}-5000$$

Steps for Completing the Square

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.

$$0.0001x^{2}+x-192=0$$

Add 192 to both sides of the equation.

$$0.0001x^{2}+x-192-\left(-192\right)=-\left(-192\right)$$

Subtracting -192 from itself leaves 0.

$$0.0001x^{2}+x=-\left(-192\right)$$

Subtract -192 from 0.

$$0.0001x^{2}+x=192$$

Multiply both sides by 10000.

$$\frac{0.0001x^{2}+x}{0.0001}=\frac{192}{0.0001}$$

Dividing by 0.0001 undoes the multiplication by 0.0001.

$$x^{2}+\frac{1}{0.0001}x=\frac{192}{0.0001}$$

Divide 1 by 0.0001 by multiplying 1 by the reciprocal of 0.0001.

$$x^{2}+10000x=\frac{192}{0.0001}$$

Divide 192 by 0.0001 by multiplying 192 by the reciprocal of 0.0001.

$$x^{2}+10000x=1920000$$

Divide 10000, the coefficient of the x term, by 2 to get 5000. Then add the square of 5000 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+10000x+5000^{2}=1920000+5000^{2}$$

Square 5000.

$$x^{2}+10000x+25000000=1920000+25000000$$

Add 1920000 to 25000000.

$$x^{2}+10000x+25000000=26920000$$

Factor x^{2}+10000x+25000000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+5000\right)^{2}=26920000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+5000\right)^{2}}=\sqrt{26920000}$$

Simplify.

$$x+5000=200\sqrt{673} x+5000=-200\sqrt{673}$$

Subtract 5000 from both sides of the equation.

$$x=200\sqrt{673}-5000 x=-200\sqrt{673}-5000$$