Solve 1/51/3x^2-25x+36=0

$$\frac { 1 } { 5 } \frac { 1 } { 3 } x ^ { 2 } - 25 x + 36 = 0$$

Solve for x

$$x = \frac{3 \sqrt{15385} + 375}{2} \approx 373.554427521$$

$$x = \frac{375 - 3 \sqrt{15385}}{2} \approx 1.445572479$$

Steps Using the Quadratic Formula

Multiply \frac{1}{5} and \frac{1}{3} to get \frac{1}{15}.

$$\frac{1}{15}x^{2}-25x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{15} for a, -25 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times \frac{1}{15}\times 36}}{2\times \frac{1}{15}}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times \frac{1}{15}\times 36}}{2\times \frac{1}{15}}$$

Multiply -4 times \frac{1}{15}.

$$x=\frac{-\left(-25\right)±\sqrt{625-\frac{4}{15}\times 36}}{2\times \frac{1}{15}}$$

Multiply -\frac{4}{15} times 36.

$$x=\frac{-\left(-25\right)±\sqrt{625-\frac{48}{5}}}{2\times \frac{1}{15}}$$

Add 625 to -\frac{48}{5}.

$$x=\frac{-\left(-25\right)±\sqrt{\frac{3077}{5}}}{2\times \frac{1}{15}}$$

Take the square root of \frac{3077}{5}.

$$x=\frac{-\left(-25\right)±\frac{\sqrt{15385}}{5}}{2\times \frac{1}{15}}$$

The opposite of -25 is 25.

$$x=\frac{25±\frac{\sqrt{15385}}{5}}{2\times \frac{1}{15}}$$

Multiply 2 times \frac{1}{15}.

$$x=\frac{25±\frac{\sqrt{15385}}{5}}{\frac{2}{15}}$$

Now solve the equation x=\frac{25±\frac{\sqrt{15385}}{5}}{\frac{2}{15}} when ± is plus. Add 25 to \frac{\sqrt{15385}}{5}.

$$x=\frac{\frac{\sqrt{15385}}{5}+25}{\frac{2}{15}}$$

Divide 25+\frac{\sqrt{15385}}{5} by \frac{2}{15} by multiplying 25+\frac{\sqrt{15385}}{5} by the reciprocal of \frac{2}{15}.

$$x=\frac{3\sqrt{15385}+375}{2}$$

Now solve the equation x=\frac{25±\frac{\sqrt{15385}}{5}}{\frac{2}{15}} when ± is minus. Subtract \frac{\sqrt{15385}}{5} from 25.

$$x=\frac{-\frac{\sqrt{15385}}{5}+25}{\frac{2}{15}}$$

Divide 25-\frac{\sqrt{15385}}{5} by \frac{2}{15} by multiplying 25-\frac{\sqrt{15385}}{5} by the reciprocal of \frac{2}{15}.

$$x=\frac{375-3\sqrt{15385}}{2}$$

The equation is now solved.

$$x=\frac{3\sqrt{15385}+375}{2} x=\frac{375-3\sqrt{15385}}{2}$$

Steps for Completing the Square

Multiply \frac{1}{5} and \frac{1}{3} to get \frac{1}{15}.

$$\frac{1}{15}x^{2}-25x+36=0$$

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$\frac{1}{15}x^{2}-25x=-36$$

Multiply both sides by 15.

$$\frac{\frac{1}{15}x^{2}-25x}{\frac{1}{15}}=-\frac{36}{\frac{1}{15}}$$

Dividing by \frac{1}{15} undoes the multiplication by \frac{1}{15}.

$$x^{2}+\left(-\frac{25}{\frac{1}{15}}\right)x=-\frac{36}{\frac{1}{15}}$$

Divide -25 by \frac{1}{15} by multiplying -25 by the reciprocal of \frac{1}{15}.

$$x^{2}-375x=-\frac{36}{\frac{1}{15}}$$

Divide -36 by \frac{1}{15} by multiplying -36 by the reciprocal of \frac{1}{15}.

$$x^{2}-375x=-540$$

Divide -375, the coefficient of the x term, by 2 to get -\frac{375}{2}. Then add the square of -\frac{375}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-375x+\left(-\frac{375}{2}\right)^{2}=-540+\left(-\frac{375}{2}\right)^{2}$$

Square -\frac{375}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-375x+\frac{140625}{4}=-540+\frac{140625}{4}$$

Add -540 to \frac{140625}{4}.

$$x^{2}-375x+\frac{140625}{4}=\frac{138465}{4}$$

Factor x^{2}-375x+\frac{140625}{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}.

$$\left(x-\frac{375}{2}\right)^{2}=\frac{138465}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{375}{2}\right)^{2}}=\sqrt{\frac{138465}{4}}$$

Simplify.

$$x-\frac{375}{2}=\frac{3\sqrt{15385}}{2} x-\frac{375}{2}=-\frac{3\sqrt{15385}}{2}$$

Add \frac{375}{2} to both sides of the equation.

$$x=\frac{3\sqrt{15385}+375}{2} x=\frac{375-3\sqrt{15385}}{2}$$