Solve x^2-6x+81=(9-3x)(2x+3)

$$x ^ { 2 } - 6 x + 81 = ( 9 - 3 x ) ( 2 x + 3 )$$

Solve for x (complex solution)

$$x=\frac{15+3\sqrt{143}i}{14}\approx 1.071428571+2.562484445i$$

$$x=\frac{-3\sqrt{143}i+15}{14}\approx 1.071428571-2.562484445i$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 9-3x by 2x+3 and combine like terms.

$$x^{2}-6x+81=9x+27-6x^{2}$$

Subtract 9x from both sides.

$$x^{2}-6x+81-9x=27-6x^{2}$$

Combine -6x and -9x to get -15x.

$$x^{2}-15x+81=27-6x^{2}$$

Subtract 27 from both sides.

$$x^{2}-15x+81-27=-6x^{2}$$

Subtract 27 from 81 to get 54.

$$x^{2}-15x+54=-6x^{2}$$

Add 6x^{2} to both sides.

$$x^{2}-15x+54+6x^{2}=0$$

Combine x^{2} and 6x^{2} to get 7x^{2}.

$$7x^{2}-15x+54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 7 for a, -15 for b, and 54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\times 7\times 54}}{2\times 7}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\times 7\times 54}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-15\right)±\sqrt{225-28\times 54}}{2\times 7}$$

Multiply -28 times 54.

$$x=\frac{-\left(-15\right)±\sqrt{225-1512}}{2\times 7}$$

Add 225 to -1512.

$$x=\frac{-\left(-15\right)±\sqrt{-1287}}{2\times 7}$$

Take the square root of -1287.

$$x=\frac{-\left(-15\right)±3\sqrt{143}i}{2\times 7}$$

The opposite of -15 is 15.

$$x=\frac{15±3\sqrt{143}i}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{15±3\sqrt{143}i}{14}$$

Now solve the equation x=\frac{15±3\sqrt{143}i}{14} when ± is plus. Add 15 to 3i\sqrt{143}.

$$x=\frac{15+3\sqrt{143}i}{14}$$

Now solve the equation x=\frac{15±3\sqrt{143}i}{14} when ± is minus. Subtract 3i\sqrt{143} from 15.

$$x=\frac{-3\sqrt{143}i+15}{14}$$

The equation is now solved.

$$x=\frac{15+3\sqrt{143}i}{14} x=\frac{-3\sqrt{143}i+15}{14}$$

Steps for Completing the Square

Use the distributive property to multiply 9-3x by 2x+3 and combine like terms.

$$x^{2}-6x+81=9x+27-6x^{2}$$

Subtract 9x from both sides.

$$x^{2}-6x+81-9x=27-6x^{2}$$

Combine -6x and -9x to get -15x.

$$x^{2}-15x+81=27-6x^{2}$$

Add 6x^{2} to both sides.

$$x^{2}-15x+81+6x^{2}=27$$

Combine x^{2} and 6x^{2} to get 7x^{2}.

$$7x^{2}-15x+81=27$$

Subtract 81 from both sides.

$$7x^{2}-15x=27-81$$

Subtract 81 from 27 to get -54.

$$7x^{2}-15x=-54$$

Divide both sides by 7.

$$\frac{7x^{2}-15x}{7}=-\frac{54}{7}$$

Dividing by 7 undoes the multiplication by 7.

$$x^{2}-\frac{15}{7}x=-\frac{54}{7}$$

Divide -\frac{15}{7}, the coefficient of the x term, by 2 to get -\frac{15}{14}. Then add the square of -\frac{15}{14} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{15}{7}x+\left(-\frac{15}{14}\right)^{2}=-\frac{54}{7}+\left(-\frac{15}{14}\right)^{2}$$

Square -\frac{15}{14} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{15}{7}x+\frac{225}{196}=-\frac{54}{7}+\frac{225}{196}$$

Add -\frac{54}{7} to \frac{225}{196} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{15}{7}x+\frac{225}{196}=-\frac{1287}{196}$$

Factor x^{2}-\frac{15}{7}x+\frac{225}{196}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{15}{14}\right)^{2}=-\frac{1287}{196}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{15}{14}\right)^{2}}=\sqrt{-\frac{1287}{196}}$$

Simplify.

$$x-\frac{15}{14}=\frac{3\sqrt{143}i}{14} x-\frac{15}{14}=-\frac{3\sqrt{143}i}{14}$$

Add \frac{15}{14} to both sides of the equation.

$$x=\frac{15+3\sqrt{143}i}{14} x=\frac{-3\sqrt{143}i+15}{14}$$