Solve 1800x^2+62000000x+600000000=0

$$1800 { x }^{ 2 } [phone]x+600000000 = 0$$

Solve for x

$$x=\frac{13000\sqrt{142}-155000}{9}\approx -9.680139826$$

$$x=\frac{-13000\sqrt{142}-155000}{9}\approx -34434.764304619$$

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.

$$1800x^{2}[phone]x+600000000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1800 for a, 62000000 for b, and 600000000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-62000000±\sqrt{62000000^{2}-4\times 1800\times 600000000}}{2\times 1800}$$

Square 62000000.

$$x=\frac{-62000000±\sqrt{3844000000000000-4\times 1800\times 600000000}}{2\times 1800}$$

Multiply -4 times 1800.

$$x=\frac{-62000000±\sqrt{3844000000000000-7200\times 600000000}}{2\times 1800}$$

Multiply -7200 times 600000000.

$$x=\frac{-62000000±\sqrt{3844000000000000-4320000000000}}{2\times 1800}$$

Add 3844000000000000 to -4320000000000.

$$x=\frac{-62000000±\sqrt{3839680000000000}}{2\times 1800}$$

Take the square root of 3839680000000000.

$$x=\frac{-62000000±5200000\sqrt{142}}{2\times 1800}$$

Multiply 2 times 1800.

$$x=\frac{-62000000±5200000\sqrt{142}}{3600}$$

Now solve the equation x=\frac{-62000000±5200000\sqrt{142}}{3600} when ± is plus. Add -62000000 to 5200000\sqrt{142}.

$$x=\frac{5200000\sqrt{142}-62000000}{3600}$$

Divide -62000000+5200000\sqrt{142} by 3600.

$$x=\frac{13000\sqrt{142}-155000}{9}$$

Now solve the equation x=\frac{-62000000±5200000\sqrt{142}}{3600} when ± is minus. Subtract 5200000\sqrt{142} from -62000000.

$$x=\frac{-5200000\sqrt{142}-62000000}{3600}$$

Divide -62000000-5200000\sqrt{142} by 3600.

$$x=\frac{-13000\sqrt{142}-155000}{9}$$

The equation is now solved.

$$x=\frac{13000\sqrt{142}-155000}{9} x=\frac{-13000\sqrt{142}-155000}{9}$$

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.

$$1800x^{2}[phone]x+600000000=0$$

Subtract 600000000 from both sides of the equation.

$$1800x^{2}[phone]x+600000000-600000000=-600000000$$

Subtracting 600000000 from itself leaves 0.

$$1800x^{2}[phone]x=-600000000$$

Divide both sides by 1800.

$$\frac{1800x^{2}[phone]x}{1800}=-\frac{600000000}{1800}$$

Dividing by 1800 undoes the multiplication by 1800.

$$x^{2}+\frac{62000000}{1800}x=-\frac{600000000}{1800}$$

Reduce the fraction \frac{62000000}{1800} to lowest terms by extracting and canceling out 200.

$$x^{2}+\frac{310000}{9}x=-\frac{600000000}{1800}$$

Reduce the fraction \frac{-600000000}{1800} to lowest terms by extracting and canceling out 600.

$$x^{2}+\frac{310000}{9}x=-\frac{1000000}{3}$$

Divide \frac{310000}{9}, the coefficient of the x term, by 2 to get \frac{155000}{9}. Then add the square of \frac{155000}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{310000}{9}x+\left(\frac{155000}{9}\right)^{2}=-\frac{1000000}{3}+\left(\frac{155000}{9}\right)^{2}$$

Square \frac{155000}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{310000}{9}x+\frac{24025000000}{81}=-\frac{1000000}{3}+\frac{24025000000}{81}$$

Add -\frac{1000000}{3} to \frac{24025000000}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{310000}{9}x+\frac{24025000000}{81}=\frac{23998000000}{81}$$

Factor x^{2}+\frac{310000}{9}x+\frac{24025000000}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{155000}{9}\right)^{2}=\frac{23998000000}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{155000}{9}\right)^{2}}=\sqrt{\frac{23998000000}{81}}$$

Simplify.

$$x+\frac{155000}{9}=\frac{13000\sqrt{142}}{9} x+\frac{155000}{9}=-\frac{13000\sqrt{142}}{9}$$

Subtract \frac{155000}{9} from both sides of the equation.

$$x=\frac{13000\sqrt{142}-155000}{9} x=\frac{-13000\sqrt{142}-155000}{9}$$