Solve 9x^2+83=54x

$$9 x ^ { 2 } + 83 = 54 x$$

Solve for x (complex solution)

$$x=\frac{\sqrt{2}i}{3}+3\approx 3+0.471404521i$$

$$x=-\frac{\sqrt{2}i}{3}+3\approx 3-0.471404521i$$

Steps Using the Quadratic Formula

Subtract 54x from both sides.

$$9x^{2}+83-54x=0$$

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.

$$9x^{2}-54x+83=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -54 for b, and 83 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 9\times 83}}{2\times 9}$$

Square -54.

$$x=\frac{-\left(-54\right)±\sqrt{2916-4\times 9\times 83}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-54\right)±\sqrt{2916-36\times 83}}{2\times 9}$$

Multiply -36 times 83.

$$x=\frac{-\left(-54\right)±\sqrt{2916-2988}}{2\times 9}$$

Add 2916 to -2988.

$$x=\frac{-\left(-54\right)±\sqrt{-72}}{2\times 9}$$

Take the square root of -72.

$$x=\frac{-\left(-54\right)±6\sqrt{2}i}{2\times 9}$$

The opposite of -54 is 54.

$$x=\frac{54±6\sqrt{2}i}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{54±6\sqrt{2}i}{18}$$

Now solve the equation x=\frac{54±6\sqrt{2}i}{18} when ± is plus. Add 54 to 6i\sqrt{2}.

$$x=\frac{54+6\sqrt{2}i}{18}$$

Divide 54+6i\sqrt{2} by 18.

$$x=\frac{\sqrt{2}i}{3}+3$$

Now solve the equation x=\frac{54±6\sqrt{2}i}{18} when ± is minus. Subtract 6i\sqrt{2} from 54.

$$x=\frac{-6\sqrt{2}i+54}{18}$$

Divide 54-6i\sqrt{2} by 18.

$$x=-\frac{\sqrt{2}i}{3}+3$$

The equation is now solved.

$$x=\frac{\sqrt{2}i}{3}+3 x=-\frac{\sqrt{2}i}{3}+3$$

Steps for Completing the Square

Subtract 54x from both sides.

$$9x^{2}+83-54x=0$$

Subtract 83 from both sides. Anything subtracted from zero gives its negation.

$$9x^{2}-54x=-83$$

Divide both sides by 9.

$$\frac{9x^{2}-54x}{9}=-\frac{83}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{54}{9}\right)x=-\frac{83}{9}$$

Divide -54 by 9.

$$x^{2}-6x=-\frac{83}{9}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-\frac{83}{9}+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-\frac{83}{9}+9$$

Add -\frac{83}{9} to 9.

$$x^{2}-6x+9=-\frac{2}{9}$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=-\frac{2}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-\frac{2}{9}}$$

Simplify.

$$x-3=\frac{\sqrt{2}i}{3} x-3=-\frac{\sqrt{2}i}{3}$$

Add 3 to both sides of the equation.

$$x=\frac{\sqrt{2}i}{3}+3 x=-\frac{\sqrt{2}i}{3}+3$$