Solve (9x+6)(60-59x)=1008

$$( 9 x + 6 ) ( 60 - 59 x ) = 1008$$

Solve for x (complex solution)

$$x=\frac{31+\sqrt{37271}i}{177}\approx 0.175141243+1.090717437i$$

$$x=\frac{-\sqrt{37271}i+31}{177}\approx 0.175141243-1.090717437i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 9x+6 by 60-59x and combine like terms.

$$186x-531x^{2}+360=1008$$

Subtract 1008 from both sides.

$$186x-531x^{2}+360-1008=0$$

Subtract 1008 from 360 to get -648.

$$186x-531x^{2}-648=0$$

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.

$$-531x^{2}+186x-648=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -531 for a, 186 for b, and -648 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-186±\sqrt{186^{2}-4\left(-531\right)\left(-648\right)}}{2\left(-531\right)}$$

Square 186.

$$x=\frac{-186±\sqrt{34596-4\left(-531\right)\left(-648\right)}}{2\left(-531\right)}$$

Multiply -4 times -531.

$$x=\frac{-186±\sqrt{34596+2124\left(-648\right)}}{2\left(-531\right)}$$

Multiply 2124 times -648.

$$x=\frac{-186±\sqrt{34596-1376352}}{2\left(-531\right)}$$

Add 34596 to -1376352.

$$x=\frac{-186±\sqrt{-1341756}}{2\left(-531\right)}$$

Take the square root of -1341756.

$$x=\frac{-186±6\sqrt{37271}i}{2\left(-531\right)}$$

Multiply 2 times -531.

$$x=\frac{-186±6\sqrt{37271}i}{-1062}$$

Now solve the equation x=\frac{-186±6\sqrt{37271}i}{-1062} when ± is plus. Add -186 to 6i\sqrt{37271}.

$$x=\frac{-186+6\sqrt{37271}i}{-1062}$$

Divide -186+6i\sqrt{37271} by -1062.

$$x=\frac{-\sqrt{37271}i+31}{177}$$

Now solve the equation x=\frac{-186±6\sqrt{37271}i}{-1062} when ± is minus. Subtract 6i\sqrt{37271} from -186.

$$x=\frac{-6\sqrt{37271}i-186}{-1062}$$

Divide -186-6i\sqrt{37271} by -1062.

$$x=\frac{31+\sqrt{37271}i}{177}$$

The equation is now solved.

$$x=\frac{-\sqrt{37271}i+31}{177} x=\frac{31+\sqrt{37271}i}{177}$$

Steps for Completing the Square

Use the distributive property to multiply 9x+6 by 60-59x and combine like terms.

$$186x-531x^{2}+360=1008$$

Subtract 360 from both sides.

$$186x-531x^{2}=1008-360$$

Subtract 360 from 1008 to get 648.

$$186x-531x^{2}=648$$

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.

$$-531x^{2}+186x=648$$

Divide both sides by -531.

$$\frac{-531x^{2}+186x}{-531}=\frac{648}{-531}$$

Dividing by -531 undoes the multiplication by -531.

$$x^{2}+\frac{186}{-531}x=\frac{648}{-531}$$

Reduce the fraction \frac{186}{-531} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{62}{177}x=\frac{648}{-531}$$

Reduce the fraction \frac{648}{-531} to lowest terms by extracting and canceling out 9.

$$x^{2}-\frac{62}{177}x=-\frac{72}{59}$$

Divide -\frac{62}{177}, the coefficient of the x term, by 2 to get -\frac{31}{177}. Then add the square of -\frac{31}{177} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{62}{177}x+\left(-\frac{31}{177}\right)^{2}=-\frac{72}{59}+\left(-\frac{31}{177}\right)^{2}$$

Square -\frac{31}{177} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{62}{177}x+\frac{961}{31329}=-\frac{72}{59}+\frac{961}{31329}$$

Add -\frac{72}{59} to \frac{961}{31329} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{62}{177}x+\frac{961}{31329}=-\frac{37271}{31329}$$

Factor x^{2}-\frac{62}{177}x+\frac{961}{31329}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{31}{177}\right)^{2}=-\frac{37271}{31329}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{31}{177}\right)^{2}}=\sqrt{-\frac{37271}{31329}}$$

Simplify.

$$x-\frac{31}{177}=\frac{\sqrt{37271}i}{177} x-\frac{31}{177}=-\frac{\sqrt{37271}i}{177}$$

Add \frac{31}{177} to both sides of the equation.

$$x=\frac{31+\sqrt{37271}i}{177} x=\frac{-\sqrt{37271}i+31}{177}$$