Solve 2x^2-27x/8-11/3=3x/27+1/12

$$2 { x }^{ 2 } - \frac{ 27x }{ 8 } - \frac{ 11 }{ 3 } = \frac{ 3x }{ 27 } + \frac{ 1 }{ 12 }$$

Solve for x

$$x = \frac{\sqrt{218521} + 251}{288} \approx 2.494660757$$

$$x=\frac{251-\sqrt{218521}}{288}\approx -0.751605201$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 216, the least common multiple of 8,3,27,12.

$$432x^{2}-27\times 27x-792=8\times 3x+18$$

Multiply -27 and 27 to get -729.

$$432x^{2}-729x-792=8\times 3x+18$$

Multiply 8 and 3 to get 24.

$$432x^{2}-729x-792=24x+18$$

Subtract 24x from both sides.

$$432x^{2}-729x-792-24x=18$$

Combine -729x and -24x to get -753x.

$$432x^{2}-753x-792=18$$

Subtract 18 from both sides.

$$432x^{2}-753x-792-18=0$$

Subtract 18 from -792 to get -810.

$$432x^{2}-753x-810=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 432 for a, -753 for b, and -810 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-753\right)±\sqrt{\left(-753\right)^{2}-4\times 432\left(-810\right)}}{2\times 432}$$

Square -753.

$$x=\frac{-\left(-753\right)±\sqrt{567009-4\times 432\left(-810\right)}}{2\times 432}$$

Multiply -4 times 432.

$$x=\frac{-\left(-753\right)±\sqrt{567009-1728\left(-810\right)}}{2\times 432}$$

Multiply -1728 times -810.

$$x=\frac{-\left(-753\right)±\sqrt{567009+1399680}}{2\times 432}$$

Add 567009 to 1399680.

$$x=\frac{-\left(-753\right)±\sqrt{1966689}}{2\times 432}$$

Take the square root of 1966689.

$$x=\frac{-\left(-753\right)±3\sqrt{218521}}{2\times 432}$$

The opposite of -753 is 753.

$$x=\frac{753±3\sqrt{218521}}{2\times 432}$$

Multiply 2 times 432.

$$x=\frac{753±3\sqrt{218521}}{864}$$

Now solve the equation x=\frac{753±3\sqrt{218521}}{864} when ± is plus. Add 753 to 3\sqrt{218521}.

$$x=\frac{3\sqrt{218521}+753}{864}$$

Divide 753+3\sqrt{218521} by 864.

$$x=\frac{\sqrt{218521}+251}{288}$$

Now solve the equation x=\frac{753±3\sqrt{218521}}{864} when ± is minus. Subtract 3\sqrt{218521} from 753.

$$x=\frac{753-3\sqrt{218521}}{864}$$

Divide 753-3\sqrt{218521} by 864.

$$x=\frac{251-\sqrt{218521}}{288}$$

The equation is now solved.

$$x=\frac{\sqrt{218521}+251}{288} x=\frac{251-\sqrt{218521}}{288}$$

Steps for Completing the Square

Multiply both sides of the equation by 216, the least common multiple of 8,3,27,12.

$$432x^{2}-27\times 27x-792=8\times 3x+18$$

Multiply -27 and 27 to get -729.

$$432x^{2}-729x-792=8\times 3x+18$$

Multiply 8 and 3 to get 24.

$$432x^{2}-729x-792=24x+18$$

Subtract 24x from both sides.

$$432x^{2}-729x-792-24x=18$$

Combine -729x and -24x to get -753x.

$$432x^{2}-753x-792=18$$

Add 792 to both sides.

$$432x^{2}-753x=18+792$$

Add 18 and 792 to get 810.

$$432x^{2}-753x=810$$

Divide both sides by 432.

$$\frac{432x^{2}-753x}{432}=\frac{810}{432}$$

Dividing by 432 undoes the multiplication by 432.

$$x^{2}+\left(-\frac{753}{432}\right)x=\frac{810}{432}$$

Reduce the fraction \frac{-753}{432} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{251}{144}x=\frac{810}{432}$$

Reduce the fraction \frac{810}{432} to lowest terms by extracting and canceling out 54.

$$x^{2}-\frac{251}{144}x=\frac{15}{8}$$

Divide -\frac{251}{144}, the coefficient of the x term, by 2 to get -\frac{251}{288}. Then add the square of -\frac{251}{288} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{251}{144}x+\left(-\frac{251}{288}\right)^{2}=\frac{15}{8}+\left(-\frac{251}{288}\right)^{2}$$

Square -\frac{251}{288} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{251}{144}x+\frac{63001}{82944}=\frac{15}{8}+\frac{63001}{82944}$$

Add \frac{15}{8} to \frac{63001}{82944} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{251}{144}x+\frac{63001}{82944}=\frac{218521}{82944}$$

Factor x^{2}-\frac{251}{144}x+\frac{63001}{82944}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{251}{288}\right)^{2}=\frac{218521}{82944}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{251}{288}\right)^{2}}=\sqrt{\frac{218521}{82944}}$$

Simplify.

$$x-\frac{251}{288}=\frac{\sqrt{218521}}{288} x-\frac{251}{288}=-\frac{\sqrt{218521}}{288}$$

Add \frac{251}{288} to both sides of the equation.

$$x=\frac{\sqrt{218521}+251}{288} x=\frac{251-\sqrt{218521}}{288}$$