Solve 0=3x^2+2700+90x

$$0 = 3 x ^ { 2 } + 2700 + 90 x$$

Solve for x (complex solution)

$$x=-15+15\sqrt{3}i\approx -15+25.980762114i$$

$$x=-15\sqrt{3}i-15\approx -15-25.980762114i$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+2700+90x=0$$

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.

$$3x^{2}+90x+2700=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 90 for b, and 2700 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-90±\sqrt{90^{2}-4\times 3\times 2700}}{2\times 3}$$

Square 90.

$$x=\frac{-90±\sqrt{8100-4\times 3\times 2700}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-90±\sqrt{8100-12\times 2700}}{2\times 3}$$

Multiply -12 times 2700.

$$x=\frac{-90±\sqrt{8100-32400}}{2\times 3}$$

Add 8100 to -32400.

$$x=\frac{-90±\sqrt{-24300}}{2\times 3}$$

Take the square root of -24300.

$$x=\frac{-90±90\sqrt{3}i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-90±90\sqrt{3}i}{6}$$

Now solve the equation x=\frac{-90±90\sqrt{3}i}{6} when ± is plus. Add -90 to 90i\sqrt{3}.

$$x=\frac{-90+90\sqrt{3}i}{6}$$

Divide -90+90i\sqrt{3} by 6.

$$x=-15+15\sqrt{3}i$$

Now solve the equation x=\frac{-90±90\sqrt{3}i}{6} when ± is minus. Subtract 90i\sqrt{3} from -90.

$$x=\frac{-90\sqrt{3}i-90}{6}$$

Divide -90-90i\sqrt{3} by 6.

$$x=-15\sqrt{3}i-15$$

The equation is now solved.

$$x=-15+15\sqrt{3}i x=-15\sqrt{3}i-15$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$3x^{2}+2700+90x=0$$

Subtract 2700 from both sides. Anything subtracted from zero gives its negation.

$$3x^{2}+90x=-2700$$

Divide both sides by 3.

$$\frac{3x^{2}+90x}{3}=-\frac{2700}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\frac{90}{3}x=-\frac{2700}{3}$$

Divide 90 by 3.

$$x^{2}+30x=-\frac{2700}{3}$$

Divide -2700 by 3.

$$x^{2}+30x=-900$$

Divide 30, the coefficient of the x term, by 2 to get 15. Then add the square of 15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+30x+15^{2}=-900+15^{2}$$

Square 15.

$$x^{2}+30x+225=-900+225$$

Add -900 to 225.

$$x^{2}+30x+225=-675$$

Factor x^{2}+30x+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+15\right)^{2}=-675$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+15\right)^{2}}=\sqrt{-675}$$

Simplify.

$$x+15=15\sqrt{3}i x+15=-15\sqrt{3}i$$

Subtract 15 from both sides of the equation.

$$x=-15+15\sqrt{3}i x=-15\sqrt{3}i-15$$