Solve -9/r-6/r+6=1

$$- \frac { 9 } { r } - \frac { 6 } { r + 6 } = 1$$

Solve for r

$$r=-18$$

$$r=-3$$

Steps Using Factoring By Grouping

Variable r cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by r\left(r+6\right), the least common multiple of r,r+6.

$$-\left(r+6\right)\times 9-r\times 6=r\left(r+6\right)$$

Use the distributive property to multiply r+6 by 9.

$$-\left(9r+54\right)-r\times 6=r\left(r+6\right)$$

To find the opposite of 9r+54, find the opposite of each term.

$$-9r-54-r\times 6=r\left(r+6\right)$$

Use the distributive property to multiply r by r+6.

$$-9r-54-r\times 6=r^{2}+6r$$

Subtract r^{2} from both sides.

$$-9r-54-r\times 6-r^{2}=6r$$

Subtract 6r from both sides.

$$-9r-54-r\times 6-r^{2}-6r=0$$

Combine -9r and -6r to get -15r.

$$-15r-54-r\times 6-r^{2}=0$$

Multiply -1 and 6 to get -6.

$$-15r-54-6r-r^{2}=0$$

Combine -15r and -6r to get -21r.

$$-21r-54-r^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-r^{2}-21r-54=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -r^{2}+ar+br-54. To find a and b, set up a system to be solved.

$$a+b=-21 ab=-\left(-54\right)=54$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 54.

$$-1,-54 -2,-27 -3,-18 -6,-9$$

Calculate the sum for each pair.

$$-1-54=-55 -2-27=-29 -3-18=-21 -6-9=-15$$

The solution is the pair that gives sum -21.

$$a=-3 b=-18$$

Rewrite -r^{2}-21r-54 as \left(-r^{2}-3r\right)+\left(-18r-54\right).

$$\left(-r^{2}-3r\right)+\left(-18r-54\right)$$

Factor out r in the first and 18 in the second group.

$$r\left(-r-3\right)+18\left(-r-3\right)$$

Factor out common term -r-3 by using distributive property.

$$\left(-r-3\right)\left(r+18\right)$$

To find equation solutions, solve -r-3=0 and r+18=0.

$$r=-3 r=-18$$

Steps Using the Quadratic Formula

Variable r cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by r\left(r+6\right), the least common multiple of r,r+6.

$$-\left(r+6\right)\times 9-r\times 6=r\left(r+6\right)$$

Use the distributive property to multiply r+6 by 9.

$$-\left(9r+54\right)-r\times 6=r\left(r+6\right)$$

To find the opposite of 9r+54, find the opposite of each term.

$$-9r-54-r\times 6=r\left(r+6\right)$$

Use the distributive property to multiply r by r+6.

$$-9r-54-r\times 6=r^{2}+6r$$

Subtract r^{2} from both sides.

$$-9r-54-r\times 6-r^{2}=6r$$

Subtract 6r from both sides.

$$-9r-54-r\times 6-r^{2}-6r=0$$

Combine -9r and -6r to get -15r.

$$-15r-54-r\times 6-r^{2}=0$$

Multiply -1 and 6 to get -6.

$$-15r-54-6r-r^{2}=0$$

Combine -15r and -6r to get -21r.

$$-21r-54-r^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$-r^{2}-21r-54=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -21 for b, and -54 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\left(-1\right)\left(-54\right)}}{2\left(-1\right)}$$

Square -21.

$$r=\frac{-\left(-21\right)±\sqrt{441-4\left(-1\right)\left(-54\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$r=\frac{-\left(-21\right)±\sqrt{441+4\left(-54\right)}}{2\left(-1\right)}$$

Multiply 4 times -54.

$$r=\frac{-\left(-21\right)±\sqrt{441-216}}{2\left(-1\right)}$$

Add 441 to -216.

$$r=\frac{-\left(-21\right)±\sqrt{225}}{2\left(-1\right)}$$

Take the square root of 225.

$$r=\frac{-\left(-21\right)±15}{2\left(-1\right)}$$

The opposite of -21 is 21.

$$r=\frac{21±15}{2\left(-1\right)}$$

Multiply 2 times -1.

$$r=\frac{21±15}{-2}$$

Now solve the equation r=\frac{21±15}{-2} when ± is plus. Add 21 to 15.

$$r=\frac{36}{-2}$$

Divide 36 by -2.

$$r=-18$$

Now solve the equation r=\frac{21±15}{-2} when ± is minus. Subtract 15 from 21.

$$r=\frac{6}{-2}$$

Divide 6 by -2.

$$r=-3$$

The equation is now solved.

$$r=-18 r=-3$$

Steps for Completing the Square

Variable r cannot be equal to any of the values -6,0 since division by zero is not defined. Multiply both sides of the equation by r\left(r+6\right), the least common multiple of r,r+6.

$$-\left(r+6\right)\times 9-r\times 6=r\left(r+6\right)$$

Use the distributive property to multiply r+6 by 9.

$$-\left(9r+54\right)-r\times 6=r\left(r+6\right)$$

To find the opposite of 9r+54, find the opposite of each term.

$$-9r-54-r\times 6=r\left(r+6\right)$$

Use the distributive property to multiply r by r+6.

$$-9r-54-r\times 6=r^{2}+6r$$

Subtract r^{2} from both sides.

$$-9r-54-r\times 6-r^{2}=6r$$

Subtract 6r from both sides.

$$-9r-54-r\times 6-r^{2}-6r=0$$

Combine -9r and -6r to get -15r.

$$-15r-54-r\times 6-r^{2}=0$$

Add 54 to both sides. Anything plus zero gives itself.

$$-15r-r\times 6-r^{2}=54$$

Multiply -1 and 6 to get -6.

$$-15r-6r-r^{2}=54$$

Combine -15r and -6r to get -21r.

$$-21r-r^{2}=54$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$-r^{2}-21r=54$$

Divide both sides by -1.

$$\frac{-r^{2}-21r}{-1}=\frac{54}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$r^{2}+\left(-\frac{21}{-1}\right)r=\frac{54}{-1}$$

Divide -21 by -1.

$$r^{2}+21r=\frac{54}{-1}$$

Divide 54 by -1.

$$r^{2}+21r=-54$$

Divide 21, the coefficient of the x term, by 2 to get \frac{21}{2}. Then add the square of \frac{21}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+21r+\left(\frac{21}{2}\right)^{2}=-54+\left(\frac{21}{2}\right)^{2}$$

Square \frac{21}{2} by squaring both the numerator and the denominator of the fraction.

$$r^{2}+21r+\frac{441}{4}=-54+\frac{441}{4}$$

Add -54 to \frac{441}{4}.

$$r^{2}+21r+\frac{441}{4}=\frac{225}{4}$$

Factor r^{2}+21r+\frac{441}{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}.

$$\left(r+\frac{21}{2}\right)^{2}=\frac{225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+\frac{21}{2}\right)^{2}}=\sqrt{\frac{225}{4}}$$

Simplify.

$$r+\frac{21}{2}=\frac{15}{2} r+\frac{21}{2}=-\frac{15}{2}$$

Subtract \frac{21}{2} from both sides of the equation.

$$r=-3 r=-18$$