Solve 6x(5x-9)=0

$$6 x ( 5 x - 9 ) = 0$$

Solve for x

$$x = \frac{9}{5} = 1\frac{4}{5} = 1.8$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 6x by 5x-9.

$$30x^{2}-54x=0$$

Factor out x.

$$x\left(30x-54\right)=0$$

To find equation solutions, solve x=0 and 30x-54=0.

$$x=0 x=\frac{9}{5}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 6x by 5x-9.

$$30x^{2}-54x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 30 for a, -54 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}}}{2\times 30}$$

Take the square root of \left(-54\right)^{2}.

$$x=\frac{-\left(-54\right)±54}{2\times 30}$$

The opposite of -54 is 54.

$$x=\frac{54±54}{2\times 30}$$

Multiply 2 times 30.

$$x=\frac{54±54}{60}$$

Now solve the equation x=\frac{54±54}{60} when ± is plus. Add 54 to 54.

$$x=\frac{108}{60}$$

Reduce the fraction \frac{108}{60} to lowest terms by extracting and canceling out 12.

$$x=\frac{9}{5}$$

Now solve the equation x=\frac{54±54}{60} when ± is minus. Subtract 54 from 54.

$$x=\frac{0}{60}$$

Divide 0 by 60.

$$x=0$$

The equation is now solved.

$$x=\frac{9}{5} x=0$$

Steps for Completing the Square

Use the distributive property to multiply 6x by 5x-9.

$$30x^{2}-54x=0$$

Divide both sides by 30.

$$\frac{30x^{2}-54x}{30}=\frac{0}{30}$$

Dividing by 30 undoes the multiplication by 30.

$$x^{2}+\left(-\frac{54}{30}\right)x=\frac{0}{30}$$

Reduce the fraction \frac{-54}{30} to lowest terms by extracting and canceling out 6.

$$x^{2}-\frac{9}{5}x=\frac{0}{30}$$

Divide 0 by 30.

$$x^{2}-\frac{9}{5}x=0$$

Divide -\frac{9}{5}, the coefficient of the x term, by 2 to get -\frac{9}{10}. Then add the square of -\frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{9}{5}x+\left(-\frac{9}{10}\right)^{2}=\left(-\frac{9}{10}\right)^{2}$$

Square -\frac{9}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{9}{5}x+\frac{81}{100}=\frac{81}{100}$$

Factor x^{2}-\frac{9}{5}x+\frac{81}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{10}\right)^{2}=\frac{81}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{10}\right)^{2}}=\sqrt{\frac{81}{100}}$$

Simplify.

$$x-\frac{9}{10}=\frac{9}{10} x-\frac{9}{10}=-\frac{9}{10}$$

Add \frac{9}{10} to both sides of the equation.

$$x=\frac{9}{5} x=0$$