Solve n(9n+10)+4=0

$$n ( 9 n + 10 ) + 4 = 0$$

Solve for n

$$n=\frac{-5+\sqrt{11}i}{9}\approx -0.555555556+0.368513866i$$

$$n=\frac{-\sqrt{11}i-5}{9}\approx -0.555555556-0.368513866i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply n by 9n+10.

$$9n^{2}+10n+4=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 10 for b, and 4 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-10±\sqrt{10^{2}-4\times 9\times 4}}{2\times 9}$$

Square 10.

$$n=\frac{-10±\sqrt{100-4\times 9\times 4}}{2\times 9}$$

Multiply -4 times 9.

$$n=\frac{-10±\sqrt{100-36\times 4}}{2\times 9}$$

Multiply -36 times 4.

$$n=\frac{-10±\sqrt{100-144}}{2\times 9}$$

Add 100 to -144.

$$n=\frac{-10±\sqrt{-44}}{2\times 9}$$

Take the square root of -44.

$$n=\frac{-10±2\sqrt{11}i}{2\times 9}$$

Multiply 2 times 9.

$$n=\frac{-10±2\sqrt{11}i}{18}$$

Now solve the equation n=\frac{-10±2\sqrt{11}i}{18} when ± is plus. Add -10 to 2i\sqrt{11}.

$$n=\frac{-10+2\sqrt{11}i}{18}$$

Divide -10+2i\sqrt{11} by 18.

$$n=\frac{-5+\sqrt{11}i}{9}$$

Now solve the equation n=\frac{-10±2\sqrt{11}i}{18} when ± is minus. Subtract 2i\sqrt{11} from -10.

$$n=\frac{-2\sqrt{11}i-10}{18}$$

Divide -10-2i\sqrt{11} by 18.

$$n=\frac{-\sqrt{11}i-5}{9}$$

The equation is now solved.

$$n=\frac{-5+\sqrt{11}i}{9} n=\frac{-\sqrt{11}i-5}{9}$$

Steps for Completing the Square

Use the distributive property to multiply n by 9n+10.

$$9n^{2}+10n+4=0$$

Subtract 4 from both sides. Anything subtracted from zero gives its negation.

$$9n^{2}+10n=-4$$

Divide both sides by 9.

$$\frac{9n^{2}+10n}{9}=-\frac{4}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$n^{2}+\frac{10}{9}n=-\frac{4}{9}$$

Divide \frac{10}{9}, the coefficient of the x term, by 2 to get \frac{5}{9}. Then add the square of \frac{5}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{10}{9}n+\left(\frac{5}{9}\right)^{2}=-\frac{4}{9}+\left(\frac{5}{9}\right)^{2}$$

Square \frac{5}{9} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{10}{9}n+\frac{25}{81}=-\frac{4}{9}+\frac{25}{81}$$

Add -\frac{4}{9} to \frac{25}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}+\frac{10}{9}n+\frac{25}{81}=-\frac{11}{81}$$

Factor n^{2}+\frac{10}{9}n+\frac{25}{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}.

$$\left(n+\frac{5}{9}\right)^{2}=-\frac{11}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{5}{9}\right)^{2}}=\sqrt{-\frac{11}{81}}$$

Simplify.

$$n+\frac{5}{9}=\frac{\sqrt{11}i}{9} n+\frac{5}{9}=-\frac{\sqrt{11}i}{9}$$

Subtract \frac{5}{9} from both sides of the equation.

$$n=\frac{-5+\sqrt{11}i}{9} n=\frac{-\sqrt{11}i-5}{9}$$