Solve 16+1/9x^2+8/3x-5+3x=0

$$16+ \frac{ 1 }{ 9 } { x }^{ 2 } + \frac{ 8 }{ 3 } x-5+3x=0$$

Solve for x

$$x=\frac{21\sqrt{5}-51}{2}\approx -2.021286236$$

$$x=\frac{-21\sqrt{5}-51}{2}\approx -48.978713764$$

Steps Using the Quadratic Formula

Subtract 5 from 16 to get 11.

$$11+\frac{1}{9}x^{2}+\frac{8}{3}x+3x=0$$

Combine \frac{8}{3}x and 3x to get \frac{17}{3}x.

$$11+\frac{1}{9}x^{2}+\frac{17}{3}x=0$$

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.

$$\frac{1}{9}x^{2}+\frac{17}{3}x+11=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{9} for a, \frac{17}{3} for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{17}{3}±\sqrt{\left(\frac{17}{3}\right)^{2}-4\times \frac{1}{9}\times 11}}{2\times \frac{1}{9}}$$

Square \frac{17}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\frac{17}{3}±\sqrt{\frac{289}{9}-4\times \frac{1}{9}\times 11}}{2\times \frac{1}{9}}$$

Multiply -4 times \frac{1}{9}.

$$x=\frac{-\frac{17}{3}±\sqrt{\frac{289}{9}-\frac{4}{9}\times 11}}{2\times \frac{1}{9}}$$

Multiply -\frac{4}{9} times 11.

$$x=\frac{-\frac{17}{3}±\sqrt{\frac{289-44}{9}}}{2\times \frac{1}{9}}$$

Add \frac{289}{9} to -\frac{44}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\frac{17}{3}±\sqrt{\frac{245}{9}}}{2\times \frac{1}{9}}$$

Take the square root of \frac{245}{9}.

$$x=\frac{-\frac{17}{3}±\frac{7\sqrt{5}}{3}}{2\times \frac{1}{9}}$$

Multiply 2 times \frac{1}{9}.

$$x=\frac{-\frac{17}{3}±\frac{7\sqrt{5}}{3}}{\frac{2}{9}}$$

Now solve the equation x=\frac{-\frac{17}{3}±\frac{7\sqrt{5}}{3}}{\frac{2}{9}} when ± is plus. Add -\frac{17}{3} to \frac{7\sqrt{5}}{3}.

$$x=\frac{7\sqrt{5}-17}{\frac{2}{9}\times 3}$$

Divide \frac{-17+7\sqrt{5}}{3} by \frac{2}{9} by multiplying \frac{-17+7\sqrt{5}}{3} by the reciprocal of \frac{2}{9}.

$$x=\frac{21\sqrt{5}-51}{2}$$

Now solve the equation x=\frac{-\frac{17}{3}±\frac{7\sqrt{5}}{3}}{\frac{2}{9}} when ± is minus. Subtract \frac{7\sqrt{5}}{3} from -\frac{17}{3}.

$$x=\frac{-7\sqrt{5}-17}{\frac{2}{9}\times 3}$$

Divide \frac{-17-7\sqrt{5}}{3} by \frac{2}{9} by multiplying \frac{-17-7\sqrt{5}}{3} by the reciprocal of \frac{2}{9}.

$$x=\frac{-21\sqrt{5}-51}{2}$$

The equation is now solved.

$$x=\frac{21\sqrt{5}-51}{2} x=\frac{-21\sqrt{5}-51}{2}$$

Steps for Completing the Square

Subtract 5 from 16 to get 11.

$$11+\frac{1}{9}x^{2}+\frac{8}{3}x+3x=0$$

Combine \frac{8}{3}x and 3x to get \frac{17}{3}x.

$$11+\frac{1}{9}x^{2}+\frac{17}{3}x=0$$

Subtract 11 from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{9}x^{2}+\frac{17}{3}x=-11$$

Multiply both sides by 9.

$$\frac{\frac{1}{9}x^{2}+\frac{17}{3}x}{\frac{1}{9}}=-\frac{11}{\frac{1}{9}}$$

Dividing by \frac{1}{9} undoes the multiplication by \frac{1}{9}.

$$x^{2}+\frac{\frac{17}{3}}{\frac{1}{9}}x=-\frac{11}{\frac{1}{9}}$$

Divide \frac{17}{3} by \frac{1}{9} by multiplying \frac{17}{3} by the reciprocal of \frac{1}{9}.

$$x^{2}+51x=-\frac{11}{\frac{1}{9}}$$

Divide -11 by \frac{1}{9} by multiplying -11 by the reciprocal of \frac{1}{9}.

$$x^{2}+51x=-99$$

Divide 51, the coefficient of the x term, by 2 to get \frac{51}{2}. Then add the square of \frac{51}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+51x+\left(\frac{51}{2}\right)^{2}=-99+\left(\frac{51}{2}\right)^{2}$$

Square \frac{51}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+51x+\frac{2601}{4}=-99+\frac{2601}{4}$$

Add -99 to \frac{2601}{4}.

$$x^{2}+51x+\frac{2601}{4}=\frac{2205}{4}$$

Factor x^{2}+51x+\frac{2601}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{51}{2}\right)^{2}=\frac{2205}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{51}{2}\right)^{2}}=\sqrt{\frac{2205}{4}}$$

Simplify.

$$x+\frac{51}{2}=\frac{21\sqrt{5}}{2} x+\frac{51}{2}=-\frac{21\sqrt{5}}{2}$$

Subtract \frac{51}{2} from both sides of the equation.

$$x=\frac{21\sqrt{5}-51}{2} x=\frac{-21\sqrt{5}-51}{2}$$