Solve 5/6x(x-9)+3/4x(x-1)=-13

$$\frac { 5 } { 6 } x ( x - 9 ) + \frac { 3 } { 4 } x ( x - 1 ) = - 13$$

Solve for x (complex solution)

$$x=\frac{99+\sqrt{2055}i}{38}\approx 2.605263158+1.192950206i$$

$$x=\frac{-\sqrt{2055}i+99}{38}\approx 2.605263158-1.192950206i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{5}{6}x by x-9.

$$\frac{5}{6}xx+\frac{5}{6}x\left(-9\right)+\frac{3}{4}x\left(x-1\right)=-13$$

Multiply x and x to get x^{2}.

$$\frac{5}{6}x^{2}+\frac{5}{6}x\left(-9\right)+\frac{3}{4}x\left(x-1\right)=-13$$

Express \frac{5}{6}\left(-9\right) as a single fraction.

$$\frac{5}{6}x^{2}+\frac{5\left(-9\right)}{6}x+\frac{3}{4}x\left(x-1\right)=-13$$

Multiply 5 and -9 to get -45.

$$\frac{5}{6}x^{2}+\frac{-45}{6}x+\frac{3}{4}x\left(x-1\right)=-13$$

Reduce the fraction \frac{-45}{6} to lowest terms by extracting and canceling out 3.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}x\left(x-1\right)=-13$$

Use the distributive property to multiply \frac{3}{4}x by x-1.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}xx+\frac{3}{4}x\left(-1\right)=-13$$

Multiply x and x to get x^{2}.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}x^{2}+\frac{3}{4}x\left(-1\right)=-13$$

Multiply \frac{3}{4} and -1 to get -\frac{3}{4}.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}x^{2}-\frac{3}{4}x=-13$$

Combine \frac{5}{6}x^{2} and \frac{3}{4}x^{2} to get \frac{19}{12}x^{2}.

$$\frac{19}{12}x^{2}-\frac{15}{2}x-\frac{3}{4}x=-13$$

Combine -\frac{15}{2}x and -\frac{3}{4}x to get -\frac{33}{4}x.

$$\frac{19}{12}x^{2}-\frac{33}{4}x=-13$$

Add 13 to both sides.

$$\frac{19}{12}x^{2}-\frac{33}{4}x+13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{19}{12} for a, -\frac{33}{4} for b, and 13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{33}{4}\right)±\sqrt{\left(-\frac{33}{4}\right)^{2}-4\times \frac{19}{12}\times 13}}{2\times \frac{19}{12}}$$

Square -\frac{33}{4} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{33}{4}\right)±\sqrt{\frac{1089}{16}-4\times \frac{19}{12}\times 13}}{2\times \frac{19}{12}}$$

Multiply -4 times \frac{19}{12}.

$$x=\frac{-\left(-\frac{33}{4}\right)±\sqrt{\frac{1089}{16}-\frac{19}{3}\times 13}}{2\times \frac{19}{12}}$$

Multiply -\frac{19}{3} times 13.

$$x=\frac{-\left(-\frac{33}{4}\right)±\sqrt{\frac{1089}{16}-\frac{247}{3}}}{2\times \frac{19}{12}}$$

Add \frac{1089}{16} to -\frac{247}{3} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x=\frac{-\left(-\frac{33}{4}\right)±\sqrt{-\frac{685}{48}}}{2\times \frac{19}{12}}$$

Take the square root of -\frac{685}{48}.

$$x=\frac{-\left(-\frac{33}{4}\right)±\frac{\sqrt{2055}i}{12}}{2\times \frac{19}{12}}$$

The opposite of -\frac{33}{4} is \frac{33}{4}.

$$x=\frac{\frac{33}{4}±\frac{\sqrt{2055}i}{12}}{2\times \frac{19}{12}}$$

Multiply 2 times \frac{19}{12}.

$$x=\frac{\frac{33}{4}±\frac{\sqrt{2055}i}{12}}{\frac{19}{6}}$$

Now solve the equation x=\frac{\frac{33}{4}±\frac{\sqrt{2055}i}{12}}{\frac{19}{6}} when ± is plus. Add \frac{33}{4} to \frac{i\sqrt{2055}}{12}.

$$x=\frac{\frac{\sqrt{2055}i}{12}+\frac{33}{4}}{\frac{19}{6}}$$

Divide \frac{33}{4}+\frac{i\sqrt{2055}}{12} by \frac{19}{6} by multiplying \frac{33}{4}+\frac{i\sqrt{2055}}{12} by the reciprocal of \frac{19}{6}.

$$x=\frac{99+\sqrt{2055}i}{38}$$

Now solve the equation x=\frac{\frac{33}{4}±\frac{\sqrt{2055}i}{12}}{\frac{19}{6}} when ± is minus. Subtract \frac{i\sqrt{2055}}{12} from \frac{33}{4}.

$$x=\frac{-\frac{\sqrt{2055}i}{12}+\frac{33}{4}}{\frac{19}{6}}$$

Divide \frac{33}{4}-\frac{i\sqrt{2055}}{12} by \frac{19}{6} by multiplying \frac{33}{4}-\frac{i\sqrt{2055}}{12} by the reciprocal of \frac{19}{6}.

$$x=\frac{-\sqrt{2055}i+99}{38}$$

The equation is now solved.

$$x=\frac{99+\sqrt{2055}i}{38} x=\frac{-\sqrt{2055}i+99}{38}$$

Steps for Completing the Square

Use the distributive property to multiply \frac{5}{6}x by x-9.

$$\frac{5}{6}xx+\frac{5}{6}x\left(-9\right)+\frac{3}{4}x\left(x-1\right)=-13$$

Multiply x and x to get x^{2}.

$$\frac{5}{6}x^{2}+\frac{5}{6}x\left(-9\right)+\frac{3}{4}x\left(x-1\right)=-13$$

Express \frac{5}{6}\left(-9\right) as a single fraction.

$$\frac{5}{6}x^{2}+\frac{5\left(-9\right)}{6}x+\frac{3}{4}x\left(x-1\right)=-13$$

Multiply 5 and -9 to get -45.

$$\frac{5}{6}x^{2}+\frac{-45}{6}x+\frac{3}{4}x\left(x-1\right)=-13$$

Reduce the fraction \frac{-45}{6} to lowest terms by extracting and canceling out 3.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}x\left(x-1\right)=-13$$

Use the distributive property to multiply \frac{3}{4}x by x-1.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}xx+\frac{3}{4}x\left(-1\right)=-13$$

Multiply x and x to get x^{2}.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}x^{2}+\frac{3}{4}x\left(-1\right)=-13$$

Multiply \frac{3}{4} and -1 to get -\frac{3}{4}.

$$\frac{5}{6}x^{2}-\frac{15}{2}x+\frac{3}{4}x^{2}-\frac{3}{4}x=-13$$

Combine \frac{5}{6}x^{2} and \frac{3}{4}x^{2} to get \frac{19}{12}x^{2}.

$$\frac{19}{12}x^{2}-\frac{15}{2}x-\frac{3}{4}x=-13$$

Combine -\frac{15}{2}x and -\frac{3}{4}x to get -\frac{33}{4}x.

$$\frac{19}{12}x^{2}-\frac{33}{4}x=-13$$

Divide both sides of the equation by \frac{19}{12}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{19}{12}x^{2}-\frac{33}{4}x}{\frac{19}{12}}=-\frac{13}{\frac{19}{12}}$$

Dividing by \frac{19}{12} undoes the multiplication by \frac{19}{12}.

$$x^{2}+\left(-\frac{\frac{33}{4}}{\frac{19}{12}}\right)x=-\frac{13}{\frac{19}{12}}$$

Divide -\frac{33}{4} by \frac{19}{12} by multiplying -\frac{33}{4} by the reciprocal of \frac{19}{12}.

$$x^{2}-\frac{99}{19}x=-\frac{13}{\frac{19}{12}}$$

Divide -13 by \frac{19}{12} by multiplying -13 by the reciprocal of \frac{19}{12}.

$$x^{2}-\frac{99}{19}x=-\frac{156}{19}$$

Divide -\frac{99}{19}, the coefficient of the x term, by 2 to get -\frac{99}{38}. Then add the square of -\frac{99}{38} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{99}{19}x+\left(-\frac{99}{38}\right)^{2}=-\frac{156}{19}+\left(-\frac{99}{38}\right)^{2}$$

Square -\frac{99}{38} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{99}{19}x+\frac{9801}{1444}=-\frac{156}{19}+\frac{9801}{1444}$$

Add -\frac{156}{19} to \frac{9801}{1444} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{99}{19}x+\frac{9801}{1444}=-\frac{2055}{1444}$$

Factor x^{2}-\frac{99}{19}x+\frac{9801}{1444}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{99}{38}\right)^{2}=-\frac{2055}{1444}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{99}{38}\right)^{2}}=\sqrt{-\frac{2055}{1444}}$$

Simplify.

$$x-\frac{99}{38}=\frac{\sqrt{2055}i}{38} x-\frac{99}{38}=-\frac{\sqrt{2055}i}{38}$$

Add \frac{99}{38} to both sides of the equation.

$$x=\frac{99+\sqrt{2055}i}{38} x=\frac{-\sqrt{2055}i+99}{38}$$