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

$$-1800 { x }^{ 2 } -62000000x-600000000=0$$

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.

$$-1800x^{2}-62000000x-600000000=0$$

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}.

$$x=\frac{-\left(-62000000\right)±\sqrt{\left(-62000000\right)^{2}-4\left(-1800\right)\left(-600000000\right)}}{2\left(-1800\right)}$$

Square -62000000.

$$x=\frac{-\left(-62000000\right)±\sqrt{3844000000000000-4\left(-1800\right)\left(-600000000\right)}}{2\left(-1800\right)}$$

Multiply -4 times -1800.

$$x=\frac{-\left(-62000000\right)±\sqrt{3844000000000000+7200\left(-600000000\right)}}{2\left(-1800\right)}$$

Multiply 7200 times -600000000.

$$x=\frac{-\left(-62000000\right)±\sqrt{3844000000000000-4320000000000}}{2\left(-1800\right)}$$

Add 3844000000000000 to -4320000000000.

$$x=\frac{-\left(-62000000\right)±\sqrt{3839680000000000}}{2\left(-1800\right)}$$

Take the square root of 3839680000000000.

$$x=\frac{-\left(-62000000\right)±5200000\sqrt{142}}{2\left(-1800\right)}$$

The opposite of -62000000 is 62000000.

$$x=\frac{62000000±5200000\sqrt{142}}{2\left(-1800\right)}$$

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}.

$$x=\frac{5200000\sqrt{142}[phone]}{-3600}$$

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.

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

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.

$$-1800x^{2}-62000000x-600000000=0$$

Add 600000000 to both sides of the equation.

$$-1800x^{2}-62000000x-600000000-\left(-600000000\right)=-\left(-600000000\right)$$

Subtracting -600000000 from itself leaves 0.

$$-1800x^{2}-62000000x=-\left(-600000000\right)$$

Subtract -600000000 from 0.

$$-1800x^{2}-62000000x=600000000$$

Divide both sides by -1800.

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

Dividing by -1800 undoes the multiplication by -1800.

$$x^{2}+\left(-\frac{62000000}{-1800}\right)x=\frac{600000000}{-1800}$$

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.

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