Solve 250000x+0.1x^2-20x-3000=0

$$250000x+0.1 { x }^{ 2 } -20x-3000=0$$

Solve for x

$$x=200\sqrt{39056251}-1249900\approx 0.01200096$$

$$x=-200\sqrt{39056251}-1249900\approx -2499800.01200096$$

Steps Using the Quadratic Formula

Combine 250000x and -20x to get 249980x.

$$249980x+0.1x^{2}-3000=0$$

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.

$$0.1x^{2}+249980x-3000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 0.1 for a, 249980 for b, and -3000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-249980±\sqrt{249980^{2}-4\times 0.1\left(-3000\right)}}{2\times 0.1}$$

Square 249980.

$$x=\frac{-249980±\sqrt{62490000400-4\times 0.1\left(-3000\right)}}{2\times 0.1}$$

Multiply -4 times 0.1.

$$x=\frac{-249980±\sqrt{62490000400-0.4\left(-3000\right)}}{2\times 0.1}$$

Multiply -0.4 times -3000.

$$x=\frac{-249980±\sqrt{62490000400+1200}}{2\times 0.1}$$

Add 62490000400 to 1200.

$$x=\frac{-249980±\sqrt{62490001600}}{2\times 0.1}$$

Take the square root of 62490001600.

$$x=\frac{-249980±40\sqrt{39056251}}{2\times 0.1}$$

Multiply 2 times 0.1.

$$x=\frac{-249980±40\sqrt{39056251}}{0.2}$$

Now solve the equation x=\frac{-249980±40\sqrt{39056251}}{0.2} when ± is plus. Add -249980 to 40\sqrt{39056251}.

$$x=\frac{40\sqrt{39056251}-249980}{0.2}$$

Divide -249980+40\sqrt{39056251} by 0.2 by multiplying -249980+40\sqrt{39056251} by the reciprocal of 0.2.

$$x=200\sqrt{39056251}-1249900$$

Now solve the equation x=\frac{-249980±40\sqrt{39056251}}{0.2} when ± is minus. Subtract 40\sqrt{39056251} from -249980.

$$x=\frac{-40\sqrt{39056251}-249980}{0.2}$$

Divide -249980-40\sqrt{39056251} by 0.2 by multiplying -249980-40\sqrt{39056251} by the reciprocal of 0.2.

$$x=-200\sqrt{39056251}-1249900$$

The equation is now solved.

$$x=200\sqrt{39056251}-1249900 x=-200\sqrt{39056251}-1249900$$

Steps for Completing the Square

Combine 250000x and -20x to get 249980x.

$$249980x+0.1x^{2}-3000=0$$

Add 3000 to both sides. Anything plus zero gives itself.

$$249980x+0.1x^{2}=3000$$

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.

$$0.1x^{2}+249980x=3000$$

Multiply both sides by 10.

$$\frac{0.1x^{2}+249980x}{0.1}=\frac{3000}{0.1}$$

Dividing by 0.1 undoes the multiplication by 0.1.

$$x^{2}+\frac{249980}{0.1}x=\frac{3000}{0.1}$$

Divide 249980 by 0.1 by multiplying 249980 by the reciprocal of 0.1.

$$x^{2}+2499800x=\frac{3000}{0.1}$$

Divide 3000 by 0.1 by multiplying 3000 by the reciprocal of 0.1.

$$x^{2}+2499800x=30000$$

Divide 2499800, the coefficient of the x term, by 2 to get 1249900. Then add the square of 1249900 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+2499800x+1249900^{2}=30000+1249900^{2}$$

Square 1249900.

$$x^{2}+2499800x+1562250010000=30000+1562250010000$$

Add 30000 to 1562250010000.

$$x^{2}+2499800x+1562250010000=1562250040000$$

Factor x^{2}+2499800x+1562250010000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+1249900\right)^{2}=1562250040000$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1249900\right)^{2}}=\sqrt{1562250040000}$$

Simplify.

$$x+1249900=200\sqrt{39056251} x+1249900=-200\sqrt{39056251}$$

Subtract 1249900 from both sides of the equation.

$$x=200\sqrt{39056251}-1249900 x=-200\sqrt{39056251}-1249900$$