Solve (3x+1/3)-1/3(17-x-1/3)=(9x-2)(x+1)

$$( 3 x + \frac { 1 } { 3 } ) - \frac { 1 } { 3 } ( 17 - \frac { x - 1 } { 3 } ) = ( 9 x - 2 ) ( x + 1 )$$

Solve for x (complex solution)

$$x=\frac{-35+\sqrt{8819}i}{162}\approx -0.216049383+0.579688463i$$

$$x=\frac{-\sqrt{8819}i-35}{162}\approx -0.216049383-0.579688463i$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$3\left(3x+\frac{1}{3}\right)-1\left(17-\frac{x-1}{3}\right)=3\left(9x-2\right)\left(x+1\right)$$

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

$$9x+1-1\left(17-\frac{x-1}{3}\right)=3\left(9x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 3 by 9x-2.

$$9x+1-1\left(17-\frac{x-1}{3}\right)=\left(27x-6\right)\left(x+1\right)$$

Use the distributive property to multiply 27x-6 by x+1 and combine like terms.

$$9x+1-1\left(17-\frac{x-1}{3}\right)=27x^{2}+21x-6$$

Divide each term of x-1 by 3 to get \frac{1}{3}x-\frac{1}{3}.

$$9x+1-1\left(17-\left(\frac{1}{3}x-\frac{1}{3}\right)\right)=27x^{2}+21x-6$$

To find the opposite of \frac{1}{3}x-\frac{1}{3}, find the opposite of each term.

$$9x+1-1\left(17-\frac{1}{3}x+\frac{1}{3}\right)=27x^{2}+21x-6$$

Add 17 and \frac{1}{3} to get \frac{52}{3}.

$$9x+1-\left(\frac{52}{3}-\frac{1}{3}x\right)=27x^{2}+21x-6$$

To find the opposite of \frac{52}{3}-\frac{1}{3}x, find the opposite of each term.

$$9x+1-\frac{52}{3}+\frac{1}{3}x=27x^{2}+21x-6$$

Subtract \frac{52}{3} from 1 to get -\frac{49}{3}.

$$9x-\frac{49}{3}+\frac{1}{3}x=27x^{2}+21x-6$$

Combine 9x and \frac{1}{3}x to get \frac{28}{3}x.

$$\frac{28}{3}x-\frac{49}{3}=27x^{2}+21x-6$$

Subtract 27x^{2} from both sides.

$$\frac{28}{3}x-\frac{49}{3}-27x^{2}=21x-6$$

Subtract 21x from both sides.

$$\frac{28}{3}x-\frac{49}{3}-27x^{2}-21x=-6$$

Combine \frac{28}{3}x and -21x to get -\frac{35}{3}x.

$$-\frac{35}{3}x-\frac{49}{3}-27x^{2}=-6$$

Add 6 to both sides.

$$-\frac{35}{3}x-\frac{49}{3}-27x^{2}+6=0$$

Add -\frac{49}{3} and 6 to get -\frac{31}{3}.

$$-\frac{35}{3}x-\frac{31}{3}-27x^{2}=0$$

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.

$$-27x^{2}-\frac{35}{3}x-\frac{31}{3}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -27 for a, -\frac{35}{3} for b, and -\frac{31}{3} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{35}{3}\right)±\sqrt{\left(-\frac{35}{3}\right)^{2}-4\left(-27\right)\left(-\frac{31}{3}\right)}}{2\left(-27\right)}$$

Square -\frac{35}{3} by squaring both the numerator and the denominator of the fraction.

$$x=\frac{-\left(-\frac{35}{3}\right)±\sqrt{\frac{1225}{9}-4\left(-27\right)\left(-\frac{31}{3}\right)}}{2\left(-27\right)}$$

Multiply -4 times -27.

$$x=\frac{-\left(-\frac{35}{3}\right)±\sqrt{\frac{1225}{9}+108\left(-\frac{31}{3}\right)}}{2\left(-27\right)}$$

Multiply 108 times -\frac{31}{3}.

$$x=\frac{-\left(-\frac{35}{3}\right)±\sqrt{\frac{1225}{9}-1116}}{2\left(-27\right)}$$

Add \frac{1225}{9} to -1116.

$$x=\frac{-\left(-\frac{35}{3}\right)±\sqrt{-\frac{8819}{9}}}{2\left(-27\right)}$$

Take the square root of -\frac{8819}{9}.

$$x=\frac{-\left(-\frac{35}{3}\right)±\frac{\sqrt{8819}i}{3}}{2\left(-27\right)}$$

The opposite of -\frac{35}{3} is \frac{35}{3}.

$$x=\frac{\frac{35}{3}±\frac{\sqrt{8819}i}{3}}{2\left(-27\right)}$$

Multiply 2 times -27.

$$x=\frac{\frac{35}{3}±\frac{\sqrt{8819}i}{3}}{-54}$$

Now solve the equation x=\frac{\frac{35}{3}±\frac{\sqrt{8819}i}{3}}{-54} when ± is plus. Add \frac{35}{3} to \frac{i\sqrt{8819}}{3}.

$$x=\frac{35+\sqrt{8819}i}{-54\times 3}$$

Divide \frac{35+i\sqrt{8819}}{3} by -54.

$$x=\frac{-\sqrt{8819}i-35}{162}$$

Now solve the equation x=\frac{\frac{35}{3}±\frac{\sqrt{8819}i}{3}}{-54} when ± is minus. Subtract \frac{i\sqrt{8819}}{3} from \frac{35}{3}.

$$x=\frac{-\sqrt{8819}i+35}{-54\times 3}$$

Divide \frac{35-i\sqrt{8819}}{3} by -54.

$$x=\frac{-35+\sqrt{8819}i}{162}$$

The equation is now solved.

$$x=\frac{-\sqrt{8819}i-35}{162} x=\frac{-35+\sqrt{8819}i}{162}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$3\left(3x+\frac{1}{3}\right)-1\left(17-\frac{x-1}{3}\right)=3\left(9x-2\right)\left(x+1\right)$$

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

$$9x+1-1\left(17-\frac{x-1}{3}\right)=3\left(9x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 3 by 9x-2.

$$9x+1-1\left(17-\frac{x-1}{3}\right)=\left(27x-6\right)\left(x+1\right)$$

Use the distributive property to multiply 27x-6 by x+1 and combine like terms.

$$9x+1-1\left(17-\frac{x-1}{3}\right)=27x^{2}+21x-6$$

Divide each term of x-1 by 3 to get \frac{1}{3}x-\frac{1}{3}.

$$9x+1-1\left(17-\left(\frac{1}{3}x-\frac{1}{3}\right)\right)=27x^{2}+21x-6$$

To find the opposite of \frac{1}{3}x-\frac{1}{3}, find the opposite of each term.

$$9x+1-1\left(17-\frac{1}{3}x+\frac{1}{3}\right)=27x^{2}+21x-6$$

Add 17 and \frac{1}{3} to get \frac{52}{3}.

$$9x+1-\left(\frac{52}{3}-\frac{1}{3}x\right)=27x^{2}+21x-6$$

To find the opposite of \frac{52}{3}-\frac{1}{3}x, find the opposite of each term.

$$9x+1-\frac{52}{3}+\frac{1}{3}x=27x^{2}+21x-6$$

Subtract \frac{52}{3} from 1 to get -\frac{49}{3}.

$$9x-\frac{49}{3}+\frac{1}{3}x=27x^{2}+21x-6$$

Combine 9x and \frac{1}{3}x to get \frac{28}{3}x.

$$\frac{28}{3}x-\frac{49}{3}=27x^{2}+21x-6$$

Subtract 27x^{2} from both sides.

$$\frac{28}{3}x-\frac{49}{3}-27x^{2}=21x-6$$

Subtract 21x from both sides.

$$\frac{28}{3}x-\frac{49}{3}-27x^{2}-21x=-6$$

Combine \frac{28}{3}x and -21x to get -\frac{35}{3}x.

$$-\frac{35}{3}x-\frac{49}{3}-27x^{2}=-6$$

Add \frac{49}{3} to both sides.

$$-\frac{35}{3}x-27x^{2}=-6+\frac{49}{3}$$

Add -6 and \frac{49}{3} to get \frac{31}{3}.

$$-\frac{35}{3}x-27x^{2}=\frac{31}{3}$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-27x^{2}-\frac{35}{3}x=\frac{31}{3}$$

Divide both sides by -27.

$$\frac{-27x^{2}-\frac{35}{3}x}{-27}=\frac{\frac{31}{3}}{-27}$$

Dividing by -27 undoes the multiplication by -27.

$$x^{2}+\left(-\frac{\frac{35}{3}}{-27}\right)x=\frac{\frac{31}{3}}{-27}$$

Divide -\frac{35}{3} by -27.

$$x^{2}+\frac{35}{81}x=\frac{\frac{31}{3}}{-27}$$

Divide \frac{31}{3} by -27.

$$x^{2}+\frac{35}{81}x=-\frac{31}{81}$$

Divide \frac{35}{81}, the coefficient of the x term, by 2 to get \frac{35}{162}. Then add the square of \frac{35}{162} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{35}{81}x+\left(\frac{35}{162}\right)^{2}=-\frac{31}{81}+\left(\frac{35}{162}\right)^{2}$$

Square \frac{35}{162} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{35}{81}x+\frac{1225}{26244}=-\frac{31}{81}+\frac{1225}{26244}$$

Add -\frac{31}{81} to \frac{1225}{26244} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{35}{81}x+\frac{1225}{26244}=-\frac{8819}{26244}$$

Factor x^{2}+\frac{35}{81}x+\frac{1225}{26244}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{35}{162}\right)^{2}=-\frac{8819}{26244}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{35}{162}\right)^{2}}=\sqrt{-\frac{8819}{26244}}$$

Simplify.

$$x+\frac{35}{162}=\frac{\sqrt{8819}i}{162} x+\frac{35}{162}=-\frac{\sqrt{8819}i}{162}$$

Subtract \frac{35}{162} from both sides of the equation.

$$x=\frac{-35+\sqrt{8819}i}{162} x=\frac{-\sqrt{8819}i-35}{162}$$