Solve 27x^2+2160x-32400=0

$$27 { x }^{ 2 } +2160x-32400 = 0$$

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.

$$27x^{2}+2160x-32400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, 2160 for b, and -32400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2160±\sqrt{2160^{2}-4\times 27\left(-32400\right)}}{2\times 27}$$

Square 2160.

$$x=\frac{-2160±\sqrt{4665600-4\times 27\left(-32400\right)}}{2\times 27}$$

Multiply -4 times 27.

$$x=\frac{-2160±\sqrt{4665600-108\left(-32400\right)}}{2\times 27}$$

Multiply -108 times -32400.

$$x=\frac{-2160±\sqrt{4665600+3499200}}{2\times 27}$$

Add 4665600 to 3499200.

$$x=\frac{-2160±\sqrt{8164800}}{2\times 27}$$

Take the square root of 8164800.

$$x=\frac{-2160±1080\sqrt{7}}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{-2160±1080\sqrt{7}}{54}$$

Now solve the equation x=\frac{-2160±1080\sqrt{7}}{54} when ± is plus. Add -2160 to 1080\sqrt{7}.

$$x=\frac{1080\sqrt{7}-2160}{54}$$

Divide -2160+1080\sqrt{7} by 54.

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

Now solve the equation x=\frac{-2160±1080\sqrt{7}}{54} when ± is minus. Subtract 1080\sqrt{7} from -2160.

$$x=\frac{-1080\sqrt{7}-2160}{54}$$

Divide -2160-1080\sqrt{7} by 54.

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

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.

$$27x^{2}+2160x-32400=0$$

Add 32400 to both sides of the equation.

$$27x^{2}+2160x-32400-\left(-32400\right)=-\left(-32400\right)$$

Subtracting -32400 from itself leaves 0.

$$27x^{2}+2160x=-\left(-32400\right)$$

Subtract -32400 from 0.

$$27x^{2}+2160x=32400$$

Divide both sides by 27.

$$\frac{27x^{2}+2160x}{27}=\frac{32400}{27}$$

Dividing by 27 undoes the multiplication by 27.

$$x^{2}+\frac{2160}{27}x=\frac{32400}{27}$$

Divide 2160 by 27.

$$x^{2}+80x=\frac{32400}{27}$$

Divide 32400 by 27.

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

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.

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