Solve 2700x^2+216000x-3240000=0

$$2700 { x }^{ 2 } +216000x-3240000 = 0$$

Solve for x

$$x=20\sqrt{7}-40\approx 12.915026221$$

$$x=-20\sqrt{7}-40\approx -92.915026221$$

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.

$$2700x^{2}+216000x-3240000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2700 for a, 216000 for b, and -3240000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-216000±\sqrt{216000^{2}-4\times 2700\left(-3240000\right)}}{2\times 2700}$$

Square 216000.

$$x=\frac{-216000±\sqrt{46656000000-4\times 2700\left(-3240000\right)}}{2\times 2700}$$

Multiply -4 times 2700.

$$x=\frac{-216000±\sqrt{46656000000-10800\left(-3240000\right)}}{2\times 2700}$$

Multiply -10800 times -3240000.

$$x=\frac{-216000±\sqrt{46656000000+34992000000}}{2\times 2700}$$

Add 46656000000 to 34992000000.

$$x=\frac{-216000±\sqrt{81648000000}}{2\times 2700}$$

Take the square root of 81648000000.

$$x=\frac{-216000±108000\sqrt{7}}{2\times 2700}$$

Multiply 2 times 2700.

$$x=\frac{-216000±108000\sqrt{7}}{5400}$$

Now solve the equation x=\frac{-216000±108000\sqrt{7}}{5400} when ± is plus. Add -216000 to 108000\sqrt{7}.

$$x=\frac{108000\sqrt{7}-216000}{5400}$$

Divide -216000+108000\sqrt{7} by 5400.

$$x=20\sqrt{7}-40$$

Now solve the equation x=\frac{-216000±108000\sqrt{7}}{5400} when ± is minus. Subtract 108000\sqrt{7} from -216000.

$$x=\frac{-108000\sqrt{7}-216000}{5400}$$

Divide -216000-108000\sqrt{7} by 5400.

$$x=-20\sqrt{7}-40$$

The equation is now solved.

$$x=20\sqrt{7}-40 x=-20\sqrt{7}-40$$

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.

$$2700x^{2}+216000x-3240000=0$$

Add 3240000 to both sides of the equation.

$$2700x^{2}+216000x-3240000-\left(-3240000\right)=-\left(-3240000\right)$$

Subtracting -3240000 from itself leaves 0.

$$2700x^{2}+216000x=-\left(-3240000\right)$$

Subtract -3240000 from 0.

$$2700x^{2}+216000x=3240000$$

Divide both sides by 2700.

$$\frac{2700x^{2}+216000x}{2700}=\frac{3240000}{2700}$$

Dividing by 2700 undoes the multiplication by 2700.

$$x^{2}+\frac{216000}{2700}x=\frac{3240000}{2700}$$

Divide 216000 by 2700.

$$x^{2}+80x=\frac{3240000}{2700}$$

Divide 3240000 by 2700.

$$x^{2}+80x=1200$$

Divide 80, the coefficient of the x term, by 2 to get 40. Then add the square of 40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+80x+40^{2}=1200+40^{2}$$

Square 40.

$$x^{2}+80x+1600=1200+1600$$

Add 1200 to 1600.

$$x^{2}+80x+1600=2800$$

Factor x^{2}+80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+40\right)^{2}=2800$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{2800}$$

Simplify.

$$x+40=20\sqrt{7} x+40=-20\sqrt{7}$$

Subtract 40 from both sides of the equation.

$$x=20\sqrt{7}-40 x=-20\sqrt{7}-40$$