Solve r(r-1)-r-8=0

$$r ( r - 1 ) - r - 8 = 0$$

Solve for r

$$r=-2$$

$$r=4$$

Steps Using Factoring

Use the distributive property to multiply r by r-1.

$$r^{2}-r-r-8=0$$

Combine -r and -r to get -2r.

$$r^{2}-2r-8=0$$

To solve the equation, factor r^{2}-2r-8 using formula r^{2}+\left(a+b\right)r+ab=\left(r+a\right)\left(r+b\right). To find a and b, set up a system to be solved.

$$a+b=-2 ab=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=-4 b=2$$

Rewrite factored expression \left(r+a\right)\left(r+b\right) using the obtained values.

$$\left(r-4\right)\left(r+2\right)$$

To find equation solutions, solve r-4=0 and r+2=0.

$$r=4 r=-2$$

Steps Using Factoring By Grouping

Use the distributive property to multiply r by r-1.

$$r^{2}-r-r-8=0$$

Combine -r and -r to get -2r.

$$r^{2}-2r-8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as r^{2}+ar+br-8. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-8\right)=-8$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -8.

$$1,-8 2,-4$$

Calculate the sum for each pair.

$$1-8=-7 2-4=-2$$

The solution is the pair that gives sum -2.

$$a=-4 b=2$$

Rewrite r^{2}-2r-8 as \left(r^{2}-4r\right)+\left(2r-8\right).

$$\left(r^{2}-4r\right)+\left(2r-8\right)$$

Factor out r in the first and 2 in the second group.

$$r\left(r-4\right)+2\left(r-4\right)$$

Factor out common term r-4 by using distributive property.

$$\left(r-4\right)\left(r+2\right)$$

To find equation solutions, solve r-4=0 and r+2=0.

$$r=4 r=-2$$

Steps Using the Quadratic Formula

Use the distributive property to multiply r by r-1.

$$r^{2}-r-r-8=0$$

Combine -r and -r to get -2r.

$$r^{2}-2r-8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -2 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-8\right)}}{2}$$

Square -2.

$$r=\frac{-\left(-2\right)±\sqrt{4-4\left(-8\right)}}{2}$$

Multiply -4 times -8.

$$r=\frac{-\left(-2\right)±\sqrt{4+32}}{2}$$

Add 4 to 32.

$$r=\frac{-\left(-2\right)±\sqrt{36}}{2}$$

Take the square root of 36.

$$r=\frac{-\left(-2\right)±6}{2}$$

The opposite of -2 is 2.

$$r=\frac{2±6}{2}$$

Now solve the equation r=\frac{2±6}{2} when ± is plus. Add 2 to 6.

$$r=\frac{8}{2}$$

Divide 8 by 2.

$$r=4$$

Now solve the equation r=\frac{2±6}{2} when ± is minus. Subtract 6 from 2.

$$r=-\frac{4}{2}$$

Divide -4 by 2.

$$r=-2$$

The equation is now solved.

$$r=4 r=-2$$

Steps for Completing the Square

Use the distributive property to multiply r by r-1.

$$r^{2}-r-r-8=0$$

Combine -r and -r to get -2r.

$$r^{2}-2r-8=0$$

Add 8 to both sides. Anything plus zero gives itself.

$$r^{2}-2r=8$$

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}-2r+1=8+1$$

Add 8 to 1.

$$r^{2}-2r+1=9$$

Factor r^{2}-2r+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-1\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-1\right)^{2}}=\sqrt{9}$$

Simplify.

$$r-1=3 r-1=-3$$

Add 1 to both sides of the equation.

$$r=4 r=-2$$