Solve (r*(r+2))div2=84

$$( r \cdot ( r + 2 ) ) \div 2 = 84$$

Solve for r

$$r=-14$$

$$r=12$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$r\left(r+2\right)=84\times 2$$

Use the distributive property to multiply r by r+2.

$$r^{2}+2r=84\times 2$$

Multiply 84 and 2 to get 168.

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

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

Multiply both sides by 2.

$$r\left(r+2\right)=84\times 2$$

Use the distributive property to multiply r by r+2.

$$r^{2}+2r=84\times 2$$

Multiply 84 and 2 to get 168.

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

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.

$$r=12 r=-14$$