Solve 80=6r+r^2

$$80 = 6 r + r ^ { 2 }$$

Solve for r (complex solution)

$$r=\sqrt{89}-3\approx 6.433981132$$

$$r=-\left(\sqrt{89}+3\right)\approx -12.433981132$$

Steps Using the Quadratic Formula

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

$$6r+r^{2}=80$$

Subtract 80 from both sides.

$$6r+r^{2}-80=0$$

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.

$$r^{2}+6r-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-6±\sqrt{6^{2}-4\left(-80\right)}}{2}$$

Square 6.

$$r=\frac{-6±\sqrt{36-4\left(-80\right)}}{2}$$

Multiply -4 times -80.

$$r=\frac{-6±\sqrt{36+320}}{2}$$

Add 36 to 320.

$$r=\frac{-6±\sqrt{356}}{2}$$

Take the square root of 356.

$$r=\frac{-6±2\sqrt{89}}{2}$$

Now solve the equation r=\frac{-6±2\sqrt{89}}{2} when ± is plus. Add -6 to 2\sqrt{89}.

$$r=\frac{2\sqrt{89}-6}{2}$$

Divide -6+2\sqrt{89} by 2.

$$r=\sqrt{89}-3$$

Now solve the equation r=\frac{-6±2\sqrt{89}}{2} when ± is minus. Subtract 2\sqrt{89} from -6.

$$r=\frac{-2\sqrt{89}-6}{2}$$

Divide -6-2\sqrt{89} by 2.

$$r=-\sqrt{89}-3$$

The equation is now solved.

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$

Steps for Completing the Square

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

$$6r+r^{2}=80$$

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.

$$r^{2}+6r=80$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+6r+3^{2}=80+3^{2}$$

Square 3.

$$r^{2}+6r+9=80+9$$

Add 80 to 9.

$$r^{2}+6r+9=89$$

Factor r^{2}+6r+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+3\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+3\right)^{2}}=\sqrt{89}$$

Simplify.

$$r+3=\sqrt{89} r+3=-\sqrt{89}$$

Subtract 3 from both sides of the equation.

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$

Solve for r

$$r=\sqrt{89}-3\approx 6.433981132$$

$$r=-\sqrt{89}-3\approx -12.433981132$$

Steps Using the Quadratic Formula

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

$$6r+r^{2}=80$$

Subtract 80 from both sides.

$$6r+r^{2}-80=0$$

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.

$$r^{2}+6r-80=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 6 for b, and -80 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-6±\sqrt{6^{2}-4\left(-80\right)}}{2}$$

Square 6.

$$r=\frac{-6±\sqrt{36-4\left(-80\right)}}{2}$$

Multiply -4 times -80.

$$r=\frac{-6±\sqrt{36+320}}{2}$$

Add 36 to 320.

$$r=\frac{-6±\sqrt{356}}{2}$$

Take the square root of 356.

$$r=\frac{-6±2\sqrt{89}}{2}$$

Now solve the equation r=\frac{-6±2\sqrt{89}}{2} when ± is plus. Add -6 to 2\sqrt{89}.

$$r=\frac{2\sqrt{89}-6}{2}$$

Divide -6+2\sqrt{89} by 2.

$$r=\sqrt{89}-3$$

Now solve the equation r=\frac{-6±2\sqrt{89}}{2} when ± is minus. Subtract 2\sqrt{89} from -6.

$$r=\frac{-2\sqrt{89}-6}{2}$$

Divide -6-2\sqrt{89} by 2.

$$r=-\sqrt{89}-3$$

The equation is now solved.

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$

Steps for Completing the Square

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

$$6r+r^{2}=80$$

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.

$$r^{2}+6r=80$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$r^{2}+6r+3^{2}=80+3^{2}$$

Square 3.

$$r^{2}+6r+9=80+9$$

Add 80 to 9.

$$r^{2}+6r+9=89$$

Factor r^{2}+6r+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(r+3\right)^{2}=89$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r+3\right)^{2}}=\sqrt{89}$$

Simplify.

$$r+3=\sqrt{89} r+3=-\sqrt{89}$$

Subtract 3 from both sides of the equation.

$$r=\sqrt{89}-3 r=-\sqrt{89}-3$$