Solve 380=9t+1/2t^2

$$380 = 9 t + \frac { 1 } { 2 } t ^ { 2 }$$

Solve for t

$$t=-38$$

$$t=20$$

Steps Using the Quadratic Formula

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

$$9t+\frac{1}{2}t^{2}=380$$

Subtract 380 from both sides.

$$9t+\frac{1}{2}t^{2}-380=0$$

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.

$$\frac{1}{2}t^{2}+9t-380=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{1}{2} for a, 9 for b, and -380 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-9±\sqrt{9^{2}-4\times \frac{1}{2}\left(-380\right)}}{2\times \frac{1}{2}}$$

Square 9.

$$t=\frac{-9±\sqrt{81-4\times \frac{1}{2}\left(-380\right)}}{2\times \frac{1}{2}}$$

Multiply -4 times \frac{1}{2}.

$$t=\frac{-9±\sqrt{81-2\left(-380\right)}}{2\times \frac{1}{2}}$$

Multiply -2 times -380.

$$t=\frac{-9±\sqrt{81+760}}{2\times \frac{1}{2}}$$

Add 81 to 760.

$$t=\frac{-9±\sqrt{841}}{2\times \frac{1}{2}}$$

Take the square root of 841.

$$t=\frac{-9±29}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$t=\frac{-9±29}{1}$$

Now solve the equation t=\frac{-9±29}{1} when ± is plus. Add -9 to 29.

$$t=\frac{20}{1}$$

Divide 20 by 1.

$$t=20$$

Now solve the equation t=\frac{-9±29}{1} when ± is minus. Subtract 29 from -9.

$$t=-\frac{38}{1}$$

Divide -38 by 1.

$$t=-38$$

The equation is now solved.

$$t=20 t=-38$$

Steps for Completing the Square

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

$$9t+\frac{1}{2}t^{2}=380$$

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.

$$\frac{1}{2}t^{2}+9t=380$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}t^{2}+9t}{\frac{1}{2}}=\frac{380}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$t^{2}+\frac{9}{\frac{1}{2}}t=\frac{380}{\frac{1}{2}}$$

Divide 9 by \frac{1}{2} by multiplying 9 by the reciprocal of \frac{1}{2}.

$$t^{2}+18t=\frac{380}{\frac{1}{2}}$$

Divide 380 by \frac{1}{2} by multiplying 380 by the reciprocal of \frac{1}{2}.

$$t^{2}+18t=760$$

Divide 18, the coefficient of the x term, by 2 to get 9. Then add the square of 9 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+18t+9^{2}=760+9^{2}$$

Square 9.

$$t^{2}+18t+81=760+81$$

Add 760 to 81.

$$t^{2}+18t+81=841$$

Factor t^{2}+18t+81. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+9\right)^{2}=841$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+9\right)^{2}}=\sqrt{841}$$

Simplify.

$$t+9=29 t+9=-29$$

Subtract 9 from both sides of the equation.

$$t=20 t=-38$$