Solve x=3-2400x+160000x^2

$$x = 3 - 2400 x + 160000 x ^ { 2 }$$

Solve for x

$$x=\frac{\sqrt{3844801}+2401}{320000}\approx 0.013630676$$

$$x=\frac{2401-\sqrt{3844801}}{320000}\approx 0.001375574$$

Steps Using the Quadratic Formula

Subtract 3 from both sides.

$$x-3=-2400x+160000x^{2}$$

Add 2400x to both sides.

$$x-3+2400x=160000x^{2}$$

Combine x and 2400x to get 2401x.

$$2401x-3=160000x^{2}$$

Subtract 160000x^{2} from both sides.

$$2401x-3-160000x^{2}=0$$

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.

$$-160000x^{2}+2401x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -160000 for a, 2401 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2401±\sqrt{2401^{2}-4\left(-160000\right)\left(-3\right)}}{2\left(-160000\right)}$$

Square 2401.

$$x=\frac{-2401±\sqrt{5764801-4\left(-160000\right)\left(-3\right)}}{2\left(-160000\right)}$$

Multiply -4 times -160000.

$$x=\frac{-2401±\sqrt{5764801+640000\left(-3\right)}}{2\left(-160000\right)}$$

Multiply 640000 times -3.

$$x=\frac{-2401±\sqrt{5764801-1920000}}{2\left(-160000\right)}$$

Add 5764801 to -1920000.

$$x=\frac{-2401±\sqrt{3844801}}{2\left(-160000\right)}$$

Multiply 2 times -160000.

$$x=\frac{-2401±\sqrt{3844801}}{-320000}$$

Now solve the equation x=\frac{-2401±\sqrt{3844801}}{-320000} when ± is plus. Add -2401 to \sqrt{3844801}.

$$x=\frac{\sqrt{3844801}-2401}{-320000}$$

Divide -2401+\sqrt{3844801} by -320000.

$$x=\frac{2401-\sqrt{3844801}}{320000}$$

Now solve the equation x=\frac{-2401±\sqrt{3844801}}{-320000} when ± is minus. Subtract \sqrt{3844801} from -2401.

$$x=\frac{-\sqrt{3844801}-2401}{-320000}$$

Divide -2401-\sqrt{3844801} by -320000.

$$x=\frac{\sqrt{3844801}+2401}{320000}$$

The equation is now solved.

$$x=\frac{2401-\sqrt{3844801}}{320000} x=\frac{\sqrt{3844801}+2401}{320000}$$

Steps for Completing the Square

Add 2400x to both sides.

$$x+2400x=3+160000x^{2}$$

Combine x and 2400x to get 2401x.

$$2401x=3+160000x^{2}$$

Subtract 160000x^{2} from both sides.

$$2401x-160000x^{2}=3$$

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.

$$-160000x^{2}+2401x=3$$

Divide both sides by -160000.

$$\frac{-160000x^{2}+2401x}{-160000}=\frac{3}{-160000}$$

Dividing by -160000 undoes the multiplication by -160000.

$$x^{2}+\frac{2401}{-160000}x=\frac{3}{-160000}$$

Divide 2401 by -160000.

$$x^{2}-\frac{2401}{160000}x=\frac{3}{-160000}$$

Divide 3 by -160000.

$$x^{2}-\frac{2401}{160000}x=-\frac{3}{160000}$$

Divide -\frac{2401}{160000}, the coefficient of the x term, by 2 to get -\frac{2401}{320000}. Then add the square of -\frac{2401}{320000} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2401}{160000}x+\left(-\frac{2401}{320000}\right)^{2}=-\frac{3}{160000}+\left(-\frac{2401}{320000}\right)^{2}$$

Square -\frac{2401}{320000} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2401}{160000}x+\frac{5764801}{102400000000}=-\frac{3}{160000}+\frac{5764801}{102400000000}$$

Add -\frac{3}{160000} to \frac{5764801}{102400000000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2401}{160000}x+\frac{5764801}{102400000000}=\frac{3844801}{102400000000}$$

Factor x^{2}-\frac{2401}{160000}x+\frac{5764801}{102400000000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{2401}{320000}\right)^{2}=\frac{3844801}{102400000000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{2401}{320000}\right)^{2}}=\sqrt{\frac{3844801}{102400000000}}$$

Simplify.

$$x-\frac{2401}{320000}=\frac{\sqrt{3844801}}{320000} x-\frac{2401}{320000}=-\frac{\sqrt{3844801}}{320000}$$

Add \frac{2401}{320000} to both sides of the equation.

$$x=\frac{\sqrt{3844801}+2401}{320000} x=\frac{2401-\sqrt{3844801}}{320000}$$