Solve 6*left(x-3right)^2=9-6x

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

Solve for x (complex solution)

$$x=\frac{5+\sqrt{5}i}{2}\approx 2.5+1.118033989i$$

$$x=\frac{-\sqrt{5}i+5}{2}\approx 2.5-1.118033989i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

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

Use the distributive property to multiply 6 by x^{2}-6x+9.

$$6x^{2}-36x+54=9-6x$$

Subtract 9 from both sides.

$$6x^{2}-36x+54-9=-6x$$

Subtract 9 from 54 to get 45.

$$6x^{2}-36x+45=-6x$$

Add 6x to both sides.

$$6x^{2}-36x+45+6x=0$$

Combine -36x and 6x to get -30x.

$$6x^{2}-30x+45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, -30 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 6\times 45}}{2\times 6}$$

Square -30.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 6\times 45}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-30\right)±\sqrt{900-24\times 45}}{2\times 6}$$

Multiply -24 times 45.

$$x=\frac{-\left(-30\right)±\sqrt{900-1080}}{2\times 6}$$

Add 900 to -1080.

$$x=\frac{-\left(-30\right)±\sqrt{-180}}{2\times 6}$$

Take the square root of -180.

$$x=\frac{-\left(-30\right)±6\sqrt{5}i}{2\times 6}$$

The opposite of -30 is 30.

$$x=\frac{30±6\sqrt{5}i}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{30±6\sqrt{5}i}{12}$$

Now solve the equation x=\frac{30±6\sqrt{5}i}{12} when ± is plus. Add 30 to 6i\sqrt{5}.

$$x=\frac{30+6\sqrt{5}i}{12}$$

Divide 30+6i\sqrt{5} by 12.

$$x=\frac{5+\sqrt{5}i}{2}$$

Now solve the equation x=\frac{30±6\sqrt{5}i}{12} when ± is minus. Subtract 6i\sqrt{5} from 30.

$$x=\frac{-6\sqrt{5}i+30}{12}$$

Divide 30-6i\sqrt{5} by 12.

$$x=\frac{-\sqrt{5}i+5}{2}$$

The equation is now solved.

$$x=\frac{5+\sqrt{5}i}{2} x=\frac{-\sqrt{5}i+5}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-3\right)^{2}.

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

Use the distributive property to multiply 6 by x^{2}-6x+9.

$$6x^{2}-36x+54=9-6x$$

Add 6x to both sides.

$$6x^{2}-36x+54+6x=9$$

Combine -36x and 6x to get -30x.

$$6x^{2}-30x+54=9$$

Subtract 54 from both sides.

$$6x^{2}-30x=9-54$$

Subtract 54 from 9 to get -45.

$$6x^{2}-30x=-45$$

Divide both sides by 6.

$$\frac{6x^{2}-30x}{6}=-\frac{45}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\left(-\frac{30}{6}\right)x=-\frac{45}{6}$$

Divide -30 by 6.

$$x^{2}-5x=-\frac{45}{6}$$

Reduce the fraction \frac{-45}{6} to lowest terms by extracting and canceling out 3.

$$x^{2}-5x=-\frac{15}{2}$$

Divide -5, the coefficient of the x term, by 2 to get -\frac{5}{2}. Then add the square of -\frac{5}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{15}{2}+\left(-\frac{5}{2}\right)^{2}$$

Square -\frac{5}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-5x+\frac{25}{4}=-\frac{15}{2}+\frac{25}{4}$$

Add -\frac{15}{2} to \frac{25}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-5x+\frac{25}{4}=-\frac{5}{4}$$

Factor x^{2}-5x+\frac{25}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{5}{2}\right)^{2}=-\frac{5}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{-\frac{5}{4}}$$

Simplify.

$$x-\frac{5}{2}=\frac{\sqrt{5}i}{2} x-\frac{5}{2}=-\frac{\sqrt{5}i}{2}$$

Add \frac{5}{2} to both sides of the equation.

$$x=\frac{5+\sqrt{5}i}{2} x=\frac{-\sqrt{5}i+5}{2}$$