Solve 3b(9b-27)=0

$$3 b ( 9 b - 27 ) = 0$$

Solve for b

$$b=3$$

$$b=0$$

Steps Using Factoring

Use the distributive property to multiply 3b by 9b-27.

$$27b^{2}-81b=0$$

Factor out b.

$$b\left(27b-81\right)=0$$

To find equation solutions, solve b=0 and 27b-81=0.

$$b=0 b=3$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3b by 9b-27.

$$27b^{2}-81b=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 27 for a, -81 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}}}{2\times 27}$$

Take the square root of \left(-81\right)^{2}.

$$b=\frac{-\left(-81\right)±81}{2\times 27}$$

The opposite of -81 is 81.

$$b=\frac{81±81}{2\times 27}$$

Multiply 2 times 27.

$$b=\frac{81±81}{54}$$

Now solve the equation b=\frac{81±81}{54} when ± is plus. Add 81 to 81.

$$b=\frac{162}{54}$$

Divide 162 by 54.

$$b=3$$

Now solve the equation b=\frac{81±81}{54} when ± is minus. Subtract 81 from 81.

$$b=\frac{0}{54}$$

Divide 0 by 54.

$$b=0$$

The equation is now solved.

$$b=3 b=0$$

Steps for Completing the Square

Use the distributive property to multiply 3b by 9b-27.

$$27b^{2}-81b=0$$

Divide both sides by 27.

$$\frac{27b^{2}-81b}{27}=\frac{0}{27}$$

Dividing by 27 undoes the multiplication by 27.

$$b^{2}+\left(-\frac{81}{27}\right)b=\frac{0}{27}$$

Divide -81 by 27.

$$b^{2}-3b=\frac{0}{27}$$

Divide 0 by 27.

$$b^{2}-3b=0$$

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

$$b^{2}-3b+\left(-\frac{3}{2}\right)^{2}=\left(-\frac{3}{2}\right)^{2}$$

Square -\frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}-3b+\frac{9}{4}=\frac{9}{4}$$

Factor b^{2}-3b+\frac{9}{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}.

$$\left(b-\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b-\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$b-\frac{3}{2}=\frac{3}{2} b-\frac{3}{2}=-\frac{3}{2}$$

Add \frac{3}{2} to both sides of the equation.

$$b=3 b=0$$