Solve 212-1/3x^2=5/3x

$$212 - \frac { 1 } { 3 } x ^ { 2 } = \frac { 5 } { 3 } x$$

Solve for x

$$x = \frac{\sqrt{2569} - 5}{2} \approx 22.842651795$$

$$x=\frac{-\sqrt{2569}-5}{2}\approx -27.842651795$$

Steps Using the Quadratic Formula

Subtract \frac{5}{3}x from both sides.

$$212-\frac{1}{3}x^{2}-\frac{5}{3}x=0$$

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.

$$-\frac{1}{3}x^{2}-\frac{5}{3}x+212=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{1}{3} for a, -\frac{5}{3} for b, and 212 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\left(-\frac{5}{3}\right)^{2}-4\left(-\frac{1}{3}\right)\times 212}}{2\left(-\frac{1}{3}\right)}$$

Square -\frac{5}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{25}{9}-4\left(-\frac{1}{3}\right)\times 212}}{2\left(-\frac{1}{3}\right)}$$

Multiply -4 times -\frac{1}{3}.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{25}{9}+\frac{4}{3}\times 212}}{2\left(-\frac{1}{3}\right)}$$

Multiply \frac{4}{3} times 212.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{25}{9}+\frac{848}{3}}}{2\left(-\frac{1}{3}\right)}$$

Add \frac{25}{9} to \frac{848}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{5}{3}\right)±\sqrt{\frac{2569}{9}}}{2\left(-\frac{1}{3}\right)}$$

Take the square root of \frac{2569}{9}.

$$x=\frac{-\left(-\frac{5}{3}\right)±\frac{\sqrt{2569}}{3}}{2\left(-\frac{1}{3}\right)}$$

The opposite of -\frac{5}{3} is \frac{5}{3}.

$$x=\frac{\frac{5}{3}±\frac{\sqrt{2569}}{3}}{2\left(-\frac{1}{3}\right)}$$

Multiply 2 times -\frac{1}{3}.

$$x=\frac{\frac{5}{3}±\frac{\sqrt{2569}}{3}}{-\frac{2}{3}}$$

Now solve the equation x=\frac{\frac{5}{3}±\frac{\sqrt{2569}}{3}}{-\frac{2}{3}} when ± is plus. Add \frac{5}{3} to \frac{\sqrt{2569}}{3}.

$$x=\frac{\sqrt{2569}+5}{-\frac{2}{3}\times 3}$$

Divide \frac{5+\sqrt{2569}}{3} by -\frac{2}{3} by multiplying \frac{5+\sqrt{2569}}{3} by the reciprocal of -\frac{2}{3}.

$$x=\frac{-\sqrt{2569}-5}{2}$$

Now solve the equation x=\frac{\frac{5}{3}±\frac{\sqrt{2569}}{3}}{-\frac{2}{3}} when ± is minus. Subtract \frac{\sqrt{2569}}{3} from \frac{5}{3}.

$$x=\frac{5-\sqrt{2569}}{-\frac{2}{3}\times 3}$$

Divide \frac{5-\sqrt{2569}}{3} by -\frac{2}{3} by multiplying \frac{5-\sqrt{2569}}{3} by the reciprocal of -\frac{2}{3}.

$$x=\frac{\sqrt{2569}-5}{2}$$

The equation is now solved.

$$x=\frac{-\sqrt{2569}-5}{2} x=\frac{\sqrt{2569}-5}{2}$$

Steps for Completing the Square

Subtract \frac{5}{3}x from both sides.

$$212-\frac{1}{3}x^{2}-\frac{5}{3}x=0$$

Subtract 212 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{1}{3}x^{2}-\frac{5}{3}x=-212$$

Multiply both sides by -3.

$$\frac{-\frac{1}{3}x^{2}-\frac{5}{3}x}{-\frac{1}{3}}=-\frac{212}{-\frac{1}{3}}$$

Dividing by -\frac{1}{3} undoes the multiplication by -\frac{1}{3}.

$$x^{2}+\left(-\frac{\frac{5}{3}}{-\frac{1}{3}}\right)x=-\frac{212}{-\frac{1}{3}}$$

Divide -\frac{5}{3} by -\frac{1}{3} by multiplying -\frac{5}{3} by the reciprocal of -\frac{1}{3}.

$$x^{2}+5x=-\frac{212}{-\frac{1}{3}}$$

Divide -212 by -\frac{1}{3} by multiplying -212 by the reciprocal of -\frac{1}{3}.

$$x^{2}+5x=636$$

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

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=636+\left(\frac{5}{2}\right)^{2}$$

Square \frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+5x+\frac{25}{4}=636+\frac{25}{4}$$

Add 636 to \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{2569}{4}$$

Factor x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{2569}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{2569}{4}}$$

Simplify.

$$x+\frac{5}{2}=\frac{\sqrt{2569}}{2} x+\frac{5}{2}=-\frac{\sqrt{2569}}{2}$$

Subtract \frac{5}{2} from both sides of the equation.

$$x=\frac{\sqrt{2569}-5}{2} x=\frac{-\sqrt{2569}-5}{2}$$