Solve 150=-2.25left(x-6right)^2+121

$$150 = -2.25 { \left(x-6 \right) }^{ 2 } +121$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\sqrt{29}}{3}+6\approx 6-3.590109871i$$

$$x=\frac{i\times 2\sqrt{29}}{3}+6\approx 6+3.590109871i$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$150=-2.25\left(x^{2}-12x+36\right)+121$$

Use the distributive property to multiply -2.25 by x^{2}-12x+36.

$$150=-2.25x^{2}+27x-81+121$$

Add -81 and 121 to get 40.

$$150=-2.25x^{2}+27x+40$$

Swap sides so that all variable terms are on the left hand side.

$$-2.25x^{2}+27x+40=150$$

Subtract 150 from both sides.

$$-2.25x^{2}+27x+40-150=0$$

Subtract 150 from 40 to get -110.

$$-2.25x^{2}+27x-110=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -2.25 for a, 27 for b, and -110 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-2.25\right)\left(-110\right)}}{2\left(-2.25\right)}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\left(-2.25\right)\left(-110\right)}}{2\left(-2.25\right)}$$

Multiply -4 times -2.25.

$$x=\frac{-27±\sqrt{729+9\left(-110\right)}}{2\left(-2.25\right)}$$

Multiply 9 times -110.

$$x=\frac{-27±\sqrt{729-990}}{2\left(-2.25\right)}$$

Add 729 to -990.

$$x=\frac{-27±\sqrt{-261}}{2\left(-2.25\right)}$$

Take the square root of -261.

$$x=\frac{-27±3\sqrt{29}i}{2\left(-2.25\right)}$$

Multiply 2 times -2.25.

$$x=\frac{-27±3\sqrt{29}i}{-4.5}$$

Now solve the equation x=\frac{-27±3\sqrt{29}i}{-4.5} when ± is plus. Add -27 to 3i\sqrt{29}.

$$x=\frac{-27+3\sqrt{29}i}{-4.5}$$

Divide -27+3i\sqrt{29} by -4.5 by multiplying -27+3i\sqrt{29} by the reciprocal of -4.5.

$$x=-\frac{2\sqrt{29}i}{3}+6$$

Now solve the equation x=\frac{-27±3\sqrt{29}i}{-4.5} when ± is minus. Subtract 3i\sqrt{29} from -27.

$$x=\frac{-3\sqrt{29}i-27}{-4.5}$$

Divide -27-3i\sqrt{29} by -4.5 by multiplying -27-3i\sqrt{29} by the reciprocal of -4.5.

$$x=\frac{2\sqrt{29}i}{3}+6$$

The equation is now solved.

$$x=-\frac{2\sqrt{29}i}{3}+6 x=\frac{2\sqrt{29}i}{3}+6$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$150=-2.25\left(x^{2}-12x+36\right)+121$$

Use the distributive property to multiply -2.25 by x^{2}-12x+36.

$$150=-2.25x^{2}+27x-81+121$$

Add -81 and 121 to get 40.

$$150=-2.25x^{2}+27x+40$$

Swap sides so that all variable terms are on the left hand side.

$$-2.25x^{2}+27x+40=150$$

Subtract 40 from both sides.

$$-2.25x^{2}+27x=150-40$$

Subtract 40 from 150 to get 110.

$$-2.25x^{2}+27x=110$$

Divide both sides of the equation by -2.25, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-2.25x^{2}+27x}{-2.25}=\frac{110}{-2.25}$$

Dividing by -2.25 undoes the multiplication by -2.25.

$$x^{2}+\frac{27}{-2.25}x=\frac{110}{-2.25}$$

Divide 27 by -2.25 by multiplying 27 by the reciprocal of -2.25.

$$x^{2}-12x=\frac{110}{-2.25}$$

Divide 110 by -2.25 by multiplying 110 by the reciprocal of -2.25.

$$x^{2}-12x=-\frac{440}{9}$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-12x+\left(-6\right)^{2}=-\frac{440}{9}+\left(-6\right)^{2}$$

Square -6.

$$x^{2}-12x+36=-\frac{440}{9}+36$$

Add -\frac{440}{9} to 36.

$$x^{2}-12x+36=-\frac{116}{9}$$

Factor x^{2}-12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-6\right)^{2}=-\frac{116}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{-\frac{116}{9}}$$

Simplify.

$$x-6=\frac{2\sqrt{29}i}{3} x-6=-\frac{2\sqrt{29}i}{3}$$

Add 6 to both sides of the equation.

$$x=\frac{2\sqrt{29}i}{3}+6 x=-\frac{2\sqrt{29}i}{3}+6$$