Solve 3x(4x-9)=0

$$3 x ( 4 x - 9 ) = 0$$

Solve for x

$$x = \frac{9}{4} = 2\frac{1}{4} = 2.25$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 3x by 4x-9.

$$12x^{2}-27x=0$$

Factor out x.

$$x\left(12x-27\right)=0$$

To find equation solutions, solve x=0 and 12x-27=0.

$$x=0 x=\frac{9}{4}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 3x by 4x-9.

$$12x^{2}-27x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -27 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-27\right)±\sqrt{\left(-27\right)^{2}}}{2\times 12}$$

Take the square root of \left(-27\right)^{2}.

$$x=\frac{-\left(-27\right)±27}{2\times 12}$$

The opposite of -27 is 27.

$$x=\frac{27±27}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{27±27}{24}$$

Now solve the equation x=\frac{27±27}{24} when ± is plus. Add 27 to 27.

$$x=\frac{54}{24}$$

Reduce the fraction \frac{54}{24} to lowest terms by extracting and canceling out 6.

$$x=\frac{9}{4}$$

Now solve the equation x=\frac{27±27}{24} when ± is minus. Subtract 27 from 27.

$$x=\frac{0}{24}$$

Divide 0 by 24.

$$x=0$$

The equation is now solved.

$$x=\frac{9}{4} x=0$$

Steps for Completing the Square

Use the distributive property to multiply 3x by 4x-9.

$$12x^{2}-27x=0$$

Divide both sides by 12.

$$\frac{12x^{2}-27x}{12}=\frac{0}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\left(-\frac{27}{12}\right)x=\frac{0}{12}$$

Reduce the fraction \frac{-27}{12} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{9}{4}x=\frac{0}{12}$$

Divide 0 by 12.

$$x^{2}-\frac{9}{4}x=0$$

Divide -\frac{9}{4}, the coefficient of the x term, by 2 to get -\frac{9}{8}. Then add the square of -\frac{9}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{4}x+\left(-\frac{9}{8}\right)^{2}=\left(-\frac{9}{8}\right)^{2}$$

Square -\frac{9}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{4}x+\frac{81}{64}=\frac{81}{64}$$

Factor x^{2}-\frac{9}{4}x+\frac{81}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{8}\right)^{2}=\frac{81}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Simplify.

$$x-\frac{9}{8}=\frac{9}{8} x-\frac{9}{8}=-\frac{9}{8}$$

Add \frac{9}{8} to both sides of the equation.

$$x=\frac{9}{4} x=0$$