Solve 168=r^2+2r

$$168 = r ^ { 2 } + 2 r$$

Solve for r

$$r=-14$$

$$r=12$$

Steps Using Factoring

Swap sides so that all variable terms are on the left hand side.

$$r^{2}+2r=168$$

Subtract 168 from both sides.

$$r^{2}+2r-168=0$$

To solve the equation, factor r^{2}+2r-168 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.

$$a+b=2 ab=-168$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Calculate the sum for each pair.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

The solution is the pair that gives sum 2.

$$a=-12 b=14$$

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

$$\left(r-12\right)\left(r+14\right)$$

To find equation solutions, solve r-12=0 and r+14=0.

$$r=12 r=-14$$

Steps Using Factoring By Grouping

Swap sides so that all variable terms are on the left hand side.

$$r^{2}+2r=168$$

Subtract 168 from both sides.

$$r^{2}+2r-168=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as r^{2}+ar+br-168. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-168\right)=-168$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -168.

$$-1,168 -2,84 -3,56 -4,42 -6,28 -7,24 -8,21 -12,14$$

Calculate the sum for each pair.

$$-1+168=167 -2+84=82 -3+56=53 -4+42=38 -6+28=22 -7+24=17 -8+21=13 -12+14=2$$

The solution is the pair that gives sum 2.

$$a=-12 b=14$$

Rewrite r^{2}+2r-168 as \left(r^{2}-12r\right)+\left(14r-168\right).

$$\left(r^{2}-12r\right)+\left(14r-168\right)$$

Factor out r in the first and 14 in the second group.

$$r\left(r-12\right)+14\left(r-12\right)$$

Factor out common term r-12 by using distributive property.

$$\left(r-12\right)\left(r+14\right)$$

To find equation solutions, solve r-12=0 and r+14=0.

$$r=12 r=-14$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$r^{2}+2r=168$$

Subtract 168 from both sides.

$$r^{2}+2r-168=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -168 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-2±\sqrt{2^{2}-4\left(-168\right)}}{2}$$

Square 2.

$$r=\frac{-2±\sqrt{4-4\left(-168\right)}}{2}$$

Multiply -4 times -168.

$$r=\frac{-2±\sqrt{4+672}}{2}$$

Add 4 to 672.

$$r=\frac{-2±\sqrt{676}}{2}$$

Take the square root of 676.

$$r=\frac{-2±26}{2}$$

Now solve the equation r=\frac{-2±26}{2} when ± is plus. Add -2 to 26.

$$r=\frac{24}{2}$$

Divide 24 by 2.

$$r=12$$

Now solve the equation r=\frac{-2±26}{2} when ± is minus. Subtract 26 from -2.

$$r=-\frac{28}{2}$$

Divide -28 by 2.

$$r=-14$$

The equation is now solved.

$$r=12 r=-14$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$r^{2}+2r=168$$

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.

$$r^{2}+2r+1^{2}=168+1^{2}$$

Square 1.

$$r^{2}+2r+1=168+1$$

Add 168 to 1.

$$r^{2}+2r+1=169$$

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}.

$$\left(r+1\right)^{2}=169$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+1\right)^{2}}=\sqrt{169}$$

Simplify.

$$r+1=13 r+1=-13$$

Subtract 1 from both sides of the equation.

$$r=12 r=-14$$