Solve 9r(r-7)=0

$$9 r ( r - 7 ) = 0$$

Solve for r

$$r=7$$

$$r=0$$

Steps Using Factoring

Use the distributive property to multiply 9r by r-7.

$$9r^{2}-63r=0$$

Factor out r.

$$r\left(9r-63\right)=0$$

To find equation solutions, solve r=0 and 9r-63=0.

$$r=0 r=7$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 9r by r-7.

$$9r^{2}-63r=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -63 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-63\right)±\sqrt{\left(-63\right)^{2}}}{2\times 9}$$

Take the square root of \left(-63\right)^{2}.

$$r=\frac{-\left(-63\right)±63}{2\times 9}$$

The opposite of -63 is 63.

$$r=\frac{63±63}{2\times 9}$$

Multiply 2 times 9.

$$r=\frac{63±63}{18}$$

Now solve the equation r=\frac{63±63}{18} when ± is plus. Add 63 to 63.

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

Divide 126 by 18.

$$r=7$$

Now solve the equation r=\frac{63±63}{18} when ± is minus. Subtract 63 from 63.

$$r=\frac{0}{18}$$

Divide 0 by 18.

$$r=0$$

The equation is now solved.

$$r=7 r=0$$

Steps for Completing the Square

Use the distributive property to multiply 9r by r-7.

$$9r^{2}-63r=0$$

Divide both sides by 9.

$$\frac{9r^{2}-63r}{9}=\frac{0}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$r^{2}+\left(-\frac{63}{9}\right)r=\frac{0}{9}$$

Divide -63 by 9.

$$r^{2}-7r=\frac{0}{9}$$

Divide 0 by 9.

$$r^{2}-7r=0$$

Divide -7, the coefficient of the x term, by 2 to get -\frac{7}{2}. Then add the square of -\frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-7r+\left(-\frac{7}{2}\right)^{2}=\left(-\frac{7}{2}\right)^{2}$$

Square -\frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$r^{2}-7r+\frac{49}{4}=\frac{49}{4}$$

Factor r^{2}-7r+\frac{49}{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}.

$$\left(r-\frac{7}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{7}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$r-\frac{7}{2}=\frac{7}{2} r-\frac{7}{2}=-\frac{7}{2}$$

Add \frac{7}{2} to both sides of the equation.

$$r=7 r=0$$