Solve x^2-24x+144/6+(x^2-8x+10)/2=x/9-x^2+9x/18+5

$$\frac { x ^ { 2 } - 24 x + 144 } { 6 } + \frac { ( x ^ { 2 } - 8 x + 10 ) } { 2 } = \frac { x } { 9 } - \frac { x ^ { 2 } + 9 x } { 18 } + 5$$

Solve for x (complex solution)

$$x=\frac{137+\sqrt{3695}i}{26}\approx 5.269230769+2.337942756i$$

$$x=\frac{-\sqrt{3695}i+137}{26}\approx 5.269230769-2.337942756i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 18, the least common multiple of 6,2,9,18.

$$3\left(x^{2}-24x+144\right)+9\left(x^{2}-8x+10\right)=2x-\left(x^{2}+9x\right)+90$$

Use the distributive property to multiply 3 by x^{2}-24x+144.

$$3x^{2}-72x+432+9\left(x^{2}-8x+10\right)=2x-\left(x^{2}+9x\right)+90$$

Use the distributive property to multiply 9 by x^{2}-8x+10.

$$3x^{2}-72x+432+9x^{2}-72x+90=2x-\left(x^{2}+9x\right)+90$$

Combine 3x^{2} and 9x^{2} to get 12x^{2}.

$$12x^{2}-72x+432-72x+90=2x-\left(x^{2}+9x\right)+90$$

Combine -72x and -72x to get -144x.

$$12x^{2}-144x+432+90=2x-\left(x^{2}+9x\right)+90$$

Add 432 and 90 to get 522.

$$12x^{2}-144x+522=2x-\left(x^{2}+9x\right)+90$$

To find the opposite of x^{2}+9x, find the opposite of each term.

$$12x^{2}-144x+522=2x-x^{2}-9x+90$$

Combine 2x and -9x to get -7x.

$$12x^{2}-144x+522=-7x-x^{2}+90$$

Add 7x to both sides.

$$12x^{2}-144x+522+7x=-x^{2}+90$$

Combine -144x and 7x to get -137x.

$$12x^{2}-137x+522=-x^{2}+90$$

Add x^{2} to both sides.

$$12x^{2}-137x+522+x^{2}=90$$

Combine 12x^{2} and x^{2} to get 13x^{2}.

$$13x^{2}-137x+522=90$$

Subtract 90 from both sides.

$$13x^{2}-137x+522-90=0$$

Subtract 90 from 522 to get 432.

$$13x^{2}-137x+432=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -137 for b, and 432 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-137\right)±\sqrt{\left(-137\right)^{2}-4\times 13\times 432}}{2\times 13}$$

Square -137.

$$x=\frac{-\left(-137\right)±\sqrt{18769-4\times 13\times 432}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-137\right)±\sqrt{18769-52\times 432}}{2\times 13}$$

Multiply -52 times 432.

$$x=\frac{-\left(-137\right)±\sqrt{18769-22464}}{2\times 13}$$

Add 18769 to -22464.

$$x=\frac{-\left(-137\right)±\sqrt{-3695}}{2\times 13}$$

Take the square root of -3695.

$$x=\frac{-\left(-137\right)±\sqrt{3695}i}{2\times 13}$$

The opposite of -137 is 137.

$$x=\frac{137±\sqrt{3695}i}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{137±\sqrt{3695}i}{26}$$

Now solve the equation x=\frac{137±\sqrt{3695}i}{26} when ± is plus. Add 137 to i\sqrt{3695}.

$$x=\frac{137+\sqrt{3695}i}{26}$$

Now solve the equation x=\frac{137±\sqrt{3695}i}{26} when ± is minus. Subtract i\sqrt{3695} from 137.

$$x=\frac{-\sqrt{3695}i+137}{26}$$

The equation is now solved.

$$x=\frac{137+\sqrt{3695}i}{26} x=\frac{-\sqrt{3695}i+137}{26}$$

Steps for Completing the Square

Multiply both sides of the equation by 18, the least common multiple of 6,2,9,18.

$$3\left(x^{2}-24x+144\right)+9\left(x^{2}-8x+10\right)=2x-\left(x^{2}+9x\right)+90$$

Use the distributive property to multiply 3 by x^{2}-24x+144.

$$3x^{2}-72x+432+9\left(x^{2}-8x+10\right)=2x-\left(x^{2}+9x\right)+90$$

Use the distributive property to multiply 9 by x^{2}-8x+10.

$$3x^{2}-72x+432+9x^{2}-72x+90=2x-\left(x^{2}+9x\right)+90$$

Combine 3x^{2} and 9x^{2} to get 12x^{2}.

$$12x^{2}-72x+432-72x+90=2x-\left(x^{2}+9x\right)+90$$

Combine -72x and -72x to get -144x.

$$12x^{2}-144x+432+90=2x-\left(x^{2}+9x\right)+90$$

Add 432 and 90 to get 522.

$$12x^{2}-144x+522=2x-\left(x^{2}+9x\right)+90$$

To find the opposite of x^{2}+9x, find the opposite of each term.

$$12x^{2}-144x+522=2x-x^{2}-9x+90$$

Combine 2x and -9x to get -7x.

$$12x^{2}-144x+522=-7x-x^{2}+90$$

Add 7x to both sides.

$$12x^{2}-144x+522+7x=-x^{2}+90$$

Combine -144x and 7x to get -137x.

$$12x^{2}-137x+522=-x^{2}+90$$

Add x^{2} to both sides.

$$12x^{2}-137x+522+x^{2}=90$$

Combine 12x^{2} and x^{2} to get 13x^{2}.

$$13x^{2}-137x+522=90$$

Subtract 522 from both sides.

$$13x^{2}-137x=90-522$$

Subtract 522 from 90 to get -432.

$$13x^{2}-137x=-432$$

Divide both sides by 13.

$$\frac{13x^{2}-137x}{13}=-\frac{432}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}-\frac{137}{13}x=-\frac{432}{13}$$

Divide -\frac{137}{13}, the coefficient of the x term, by 2 to get -\frac{137}{26}. Then add the square of -\frac{137}{26} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{137}{13}x+\left(-\frac{137}{26}\right)^{2}=-\frac{432}{13}+\left(-\frac{137}{26}\right)^{2}$$

Square -\frac{137}{26} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{137}{13}x+\frac{18769}{676}=-\frac{432}{13}+\frac{18769}{676}$$

Add -\frac{432}{13} to \frac{18769}{676} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{137}{13}x+\frac{18769}{676}=-\frac{3695}{676}$$

Factor x^{2}-\frac{137}{13}x+\frac{18769}{676}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{137}{26}\right)^{2}=-\frac{3695}{676}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{137}{26}\right)^{2}}=\sqrt{-\frac{3695}{676}}$$

Simplify.

$$x-\frac{137}{26}=\frac{\sqrt{3695}i}{26} x-\frac{137}{26}=-\frac{\sqrt{3695}i}{26}$$

Add \frac{137}{26} to both sides of the equation.

$$x=\frac{137+\sqrt{3695}i}{26} x=\frac{-\sqrt{3695}i+137}{26}$$