Solve (3x-18)^2-5(3x-18)+6=0

$$( 3 x - 18 ) ^ { 2 } - 5 ( 3 x - 18 ) + 6 = 0$$

Solve for x

$$x = \frac{20}{3} = 6\frac{2}{3} \approx 6.666666667$$

$$x=7$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-18\right)^{2}.

$$9x^{2}-108x+324-5\left(3x-18\right)+6=0$$

Use the distributive property to multiply -5 by 3x-18.

$$9x^{2}-108x+324-15x+90+6=0$$

Combine -108x and -15x to get -123x.

$$9x^{2}-123x+324+90+6=0$$

Add 324 and 90 to get 414.

$$9x^{2}-123x+414+6=0$$

Add 414 and 6 to get 420.

$$9x^{2}-123x+420=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -123 for b, and 420 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-123\right)±\sqrt{\left(-123\right)^{2}-4\times 9\times 420}}{2\times 9}$$

Square -123.

$$x=\frac{-\left(-123\right)±\sqrt{15129-4\times 9\times 420}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-123\right)±\sqrt{15129-36\times 420}}{2\times 9}$$

Multiply -36 times 420.

$$x=\frac{-\left(-123\right)±\sqrt{15129-15120}}{2\times 9}$$

Add 15129 to -15120.

$$x=\frac{-\left(-123\right)±\sqrt{9}}{2\times 9}$$

Take the square root of 9.

$$x=\frac{-\left(-123\right)±3}{2\times 9}$$

The opposite of -123 is 123.

$$x=\frac{123±3}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{123±3}{18}$$

Now solve the equation x=\frac{123±3}{18} when ± is plus. Add 123 to 3.

$$x=\frac{126}{18}$$

Divide 126 by 18.

$$x=7$$

Now solve the equation x=\frac{123±3}{18} when ± is minus. Subtract 3 from 123.

$$x=\frac{120}{18}$$

Reduce the fraction \frac{120}{18} to lowest terms by extracting and canceling out 6.

$$x=\frac{20}{3}$$

The equation is now solved.

$$x=7 x=\frac{20}{3}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-18\right)^{2}.

$$9x^{2}-108x+324-5\left(3x-18\right)+6=0$$

Use the distributive property to multiply -5 by 3x-18.

$$9x^{2}-108x+324-15x+90+6=0$$

Combine -108x and -15x to get -123x.

$$9x^{2}-123x+324+90+6=0$$

Add 324 and 90 to get 414.

$$9x^{2}-123x+414+6=0$$

Add 414 and 6 to get 420.

$$9x^{2}-123x+420=0$$

Subtract 420 from both sides. Anything subtracted from zero gives its negation.

$$9x^{2}-123x=-420$$

Divide both sides by 9.

$$\frac{9x^{2}-123x}{9}=-\frac{420}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\left(-\frac{123}{9}\right)x=-\frac{420}{9}$$

Reduce the fraction \frac{-123}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{41}{3}x=-\frac{420}{9}$$

Reduce the fraction \frac{-420}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{41}{3}x=-\frac{140}{3}$$

Divide -\frac{41}{3}, the coefficient of the x term, by 2 to get -\frac{41}{6}. Then add the square of -\frac{41}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{41}{3}x+\left(-\frac{41}{6}\right)^{2}=-\frac{140}{3}+\left(-\frac{41}{6}\right)^{2}$$

Square -\frac{41}{6} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{41}{3}x+\frac{1681}{36}=-\frac{140}{3}+\frac{1681}{36}$$

Add -\frac{140}{3} to \frac{1681}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{41}{3}x+\frac{1681}{36}=\frac{1}{36}$$

Factor x^{2}-\frac{41}{3}x+\frac{1681}{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}.

$$\left(x-\frac{41}{6}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{41}{6}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$x-\frac{41}{6}=\frac{1}{6} x-\frac{41}{6}=-\frac{1}{6}$$

Add \frac{41}{6} to both sides of the equation.

$$x=7 x=\frac{20}{3}$$