Solve 3x^2-18x+225=6

$$3 x ^ { 2 } - 18 x + 225 = 6$$

Solve for x (complex solution)

$$x=3+8i$$

$$x=3-8i$$

Steps Using the Quadratic Formula

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.

$$3x^{2}-18x+225=6$$

Subtract 6 from both sides of the equation.

$$3x^{2}-18x+225-6=6-6$$

Subtracting 6 from itself leaves 0.

$$3x^{2}-18x+225-6=0$$

Subtract 6 from 225.

$$3x^{2}-18x+219=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, -18 for b, and 219 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\times 219}}{2\times 3}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 3\times 219}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-18\right)±\sqrt{324-12\times 219}}{2\times 3}$$

Multiply -12 times 219.

$$x=\frac{-\left(-18\right)±\sqrt{324-2628}}{2\times 3}$$

Add 324 to -2628.

$$x=\frac{-\left(-18\right)±\sqrt{-2304}}{2\times 3}$$

Take the square root of -2304.

$$x=\frac{-\left(-18\right)±48i}{2\times 3}$$

The opposite of -18 is 18.

$$x=\frac{18±48i}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{18±48i}{6}$$

Now solve the equation x=\frac{18±48i}{6} when ± is plus. Add 18 to 48i.

$$x=\frac{18+48i}{6}$$

Divide 18+48i by 6.

$$x=3+8i$$

Now solve the equation x=\frac{18±48i}{6} when ± is minus. Subtract 48i from 18.

$$x=\frac{18-48i}{6}$$

Divide 18-48i by 6.

$$x=3-8i$$

The equation is now solved.

$$x=3+8i x=3-8i$$

Steps for Completing the Square

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.

$$3x^{2}-18x+225=6$$

Subtract 225 from both sides of the equation.

$$3x^{2}-18x+225-225=6-225$$

Subtracting 225 from itself leaves 0.

$$3x^{2}-18x=6-225$$

Subtract 225 from 6.

$$3x^{2}-18x=-219$$

Divide both sides by 3.

$$\frac{3x^{2}-18x}{3}=-\frac{219}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$x^{2}+\left(-\frac{18}{3}\right)x=-\frac{219}{3}$$

Divide -18 by 3.

$$x^{2}-6x=-\frac{219}{3}$$

Divide -219 by 3.

$$x^{2}-6x=-73$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-73+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-73+9$$

Add -73 to 9.

$$x^{2}-6x+9=-64$$

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}.

$$\left(x-3\right)^{2}=-64$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-64}$$

Simplify.

$$x-3=8i x-3=-8i$$

Add 3 to both sides of the equation.

$$x=3+8i x=3-8i$$