Solve 90=19t+1/2*2t^2

$$90 = 19 t + \frac { 1 } { 2 } \cdot 2 t ^ { 2 }$$

Solve for t

$$t = \frac{\sqrt{721} - 19}{2} \approx 3.925721582$$

$$t=\frac{-\sqrt{721}-19}{2}\approx -22.925721582$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 2 to get 1.

$$90=19t+t^{2}$$

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

$$19t+t^{2}=90$$

Subtract 90 from both sides.

$$19t+t^{2}-90=0$$

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.

$$t^{2}+19t-90=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and -90 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-19±\sqrt{19^{2}-4\left(-90\right)}}{2}$$

Square 19.

$$t=\frac{-19±\sqrt{361-4\left(-90\right)}}{2}$$

Multiply -4 times -90.

$$t=\frac{-19±\sqrt{361+360}}{2}$$

Add 361 to 360.

$$t=\frac{-19±\sqrt{721}}{2}$$

Now solve the equation t=\frac{-19±\sqrt{721}}{2} when ± is plus. Add -19 to \sqrt{721}.

$$t=\frac{\sqrt{721}-19}{2}$$

Now solve the equation t=\frac{-19±\sqrt{721}}{2} when ± is minus. Subtract \sqrt{721} from -19.

$$t=\frac{-\sqrt{721}-19}{2}$$

The equation is now solved.

$$t=\frac{\sqrt{721}-19}{2} t=\frac{-\sqrt{721}-19}{2}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 2 to get 1.

$$90=19t+t^{2}$$

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

$$19t+t^{2}=90$$

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.

$$t^{2}+19t=90$$

Divide 19, the coefficient of the x term, by 2 to get \frac{19}{2}. Then add the square of \frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+19t+\left(\frac{19}{2}\right)^{2}=90+\left(\frac{19}{2}\right)^{2}$$

Square \frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+19t+\frac{361}{4}=90+\frac{361}{4}$$

Add 90 to \frac{361}{4}.

$$t^{2}+19t+\frac{361}{4}=\frac{721}{4}$$

Factor t^{2}+19t+\frac{361}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{19}{2}\right)^{2}=\frac{721}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{19}{2}\right)^{2}}=\sqrt{\frac{721}{4}}$$

Simplify.

$$t+\frac{19}{2}=\frac{\sqrt{721}}{2} t+\frac{19}{2}=-\frac{\sqrt{721}}{2}$$

Subtract \frac{19}{2} from both sides of the equation.

$$t=\frac{\sqrt{721}-19}{2} t=\frac{-\sqrt{721}-19}{2}$$