Solve 27x^2+55x+76=0

$$27 { x }^{ 2 } +55x+76=0$$

Solve for x (complex solution)

$$x=\frac{-55+\sqrt{5183}i}{54}\approx -1.018518519+1.333204726i$$

$$x=\frac{-\sqrt{5183}i-55}{54}\approx -1.018518519-1.333204726i$$

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.

$$27x^{2}+55x+76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, 55 for b, and 76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-55±\sqrt{55^{2}-4\times 27\times 76}}{2\times 27}$$

Square 55.

$$x=\frac{-55±\sqrt{3025-4\times 27\times 76}}{2\times 27}$$

Multiply -4 times 27.

$$x=\frac{-55±\sqrt{3025-108\times 76}}{2\times 27}$$

Multiply -108 times 76.

$$x=\frac{-55±\sqrt{3025-8208}}{2\times 27}$$

Add 3025 to -8208.

$$x=\frac{-55±\sqrt{-5183}}{2\times 27}$$

Take the square root of -5183.

$$x=\frac{-55±\sqrt{5183}i}{2\times 27}$$

Multiply 2 times 27.

$$x=\frac{-55±\sqrt{5183}i}{54}$$

Now solve the equation x=\frac{-55±\sqrt{5183}i}{54} when ± is plus. Add -55 to i\sqrt{5183}.

$$x=\frac{-55+\sqrt{5183}i}{54}$$

Now solve the equation x=\frac{-55±\sqrt{5183}i}{54} when ± is minus. Subtract i\sqrt{5183} from -55.

$$x=\frac{-\sqrt{5183}i-55}{54}$$

The equation is now solved.

$$x=\frac{-55+\sqrt{5183}i}{54} x=\frac{-\sqrt{5183}i-55}{54}$$

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.

$$27x^{2}+55x+76=0$$

Subtract 76 from both sides of the equation.

$$27x^{2}+55x+76-76=-76$$

Subtracting 76 from itself leaves 0.

$$27x^{2}+55x=-76$$

Divide both sides by 27.

$$\frac{27x^{2}+55x}{27}=-\frac{76}{27}$$

Dividing by 27 undoes the multiplication by 27.

$$x^{2}+\frac{55}{27}x=-\frac{76}{27}$$

Divide \frac{55}{27}, the coefficient of the x term, by 2 to get \frac{55}{54}. Then add the square of \frac{55}{54} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{55}{27}x+\left(\frac{55}{54}\right)^{2}=-\frac{76}{27}+\left(\frac{55}{54}\right)^{2}$$

Square \frac{55}{54} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{55}{27}x+\frac{3025}{2916}=-\frac{76}{27}+\frac{3025}{2916}$$

Add -\frac{76}{27} to \frac{3025}{2916} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{55}{27}x+\frac{3025}{2916}=-\frac{5183}{2916}$$

Factor x^{2}+\frac{55}{27}x+\frac{3025}{2916}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{55}{54}\right)^{2}=-\frac{5183}{2916}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{55}{54}\right)^{2}}=\sqrt{-\frac{5183}{2916}}$$

Simplify.

$$x+\frac{55}{54}=\frac{\sqrt{5183}i}{54} x+\frac{55}{54}=-\frac{\sqrt{5183}i}{54}$$

Subtract \frac{55}{54} from both sides of the equation.

$$x=\frac{-55+\sqrt{5183}i}{54} x=\frac{-\sqrt{5183}i-55}{54}$$