Solve 2.74=2.44t+1/2*1.2*t^2

$$2.74 = 2.44 t + \frac { 1 } { 2 } \times 1.2 \times t ^ { 2 }$$

Solve for t

$$t=\frac{\sqrt{7831}-61}{30}\approx 0.916431252$$

$$t=\frac{-\sqrt{7831}-61}{30}\approx -4.983097919$$

Steps Using the Quadratic Formula

Multiply \frac{1}{2} and 1.2 to get \frac{3}{5}.

$$2.74=2.44t+\frac{3}{5}t^{2}$$

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

$$2.44t+\frac{3}{5}t^{2}=2.74$$

Subtract 2.74 from both sides.

$$2.44t+\frac{3}{5}t^{2}-2.74=0$$

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.

$$\frac{3}{5}t^{2}+2.44t-2.74=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{5} for a, 2.44 for b, and -2.74 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-2.44±\sqrt{2.44^{2}-4\times \frac{3}{5}\left(-2.74\right)}}{2\times \frac{3}{5}}$$

Square 2.44 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-2.44±\sqrt{5.9536-4\times \frac{3}{5}\left(-2.74\right)}}{2\times \frac{3}{5}}$$

Multiply -4 times \frac{3}{5}.

$$t=\frac{-2.44±\sqrt{5.9536-\frac{12}{5}\left(-2.74\right)}}{2\times \frac{3}{5}}$$

Multiply -\frac{12}{5} times -2.74 by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-2.44±\sqrt{5.9536+\frac{822}{125}}}{2\times \frac{3}{5}}$$

Add 5.9536 to \frac{822}{125} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-2.44±\sqrt{\frac{7831}{625}}}{2\times \frac{3}{5}}$$

Take the square root of \frac{7831}{625}.

$$t=\frac{-2.44±\frac{\sqrt{7831}}{25}}{2\times \frac{3}{5}}$$

Multiply 2 times \frac{3}{5}.

$$t=\frac{-2.44±\frac{\sqrt{7831}}{25}}{\frac{6}{5}}$$

Now solve the equation t=\frac{-2.44±\frac{\sqrt{7831}}{25}}{\frac{6}{5}} when ± is plus. Add -2.44 to \frac{\sqrt{7831}}{25}.

$$t=\frac{\sqrt{7831}-61}{\frac{6}{5}\times 25}$$

Divide \frac{-61+\sqrt{7831}}{25} by \frac{6}{5} by multiplying \frac{-61+\sqrt{7831}}{25} by the reciprocal of \frac{6}{5}.

$$t=\frac{\sqrt{7831}-61}{30}$$

Now solve the equation t=\frac{-2.44±\frac{\sqrt{7831}}{25}}{\frac{6}{5}} when ± is minus. Subtract \frac{\sqrt{7831}}{25} from -2.44.

$$t=\frac{-\sqrt{7831}-61}{\frac{6}{5}\times 25}$$

Divide \frac{-61-\sqrt{7831}}{25} by \frac{6}{5} by multiplying \frac{-61-\sqrt{7831}}{25} by the reciprocal of \frac{6}{5}.

$$t=\frac{-\sqrt{7831}-61}{30}$$

The equation is now solved.

$$t=\frac{\sqrt{7831}-61}{30} t=\frac{-\sqrt{7831}-61}{30}$$

Steps for Completing the Square

Multiply \frac{1}{2} and 1.2 to get \frac{3}{5}.

$$2.74=2.44t+\frac{3}{5}t^{2}$$

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

$$2.44t+\frac{3}{5}t^{2}=2.74$$

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.

$$\frac{3}{5}t^{2}+2.44t=2.74$$

Divide both sides of the equation by \frac{3}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3}{5}t^{2}+2.44t}{\frac{3}{5}}=\frac{2.74}{\frac{3}{5}}$$

Dividing by \frac{3}{5} undoes the multiplication by \frac{3}{5}.

$$t^{2}+\frac{2.44}{\frac{3}{5}}t=\frac{2.74}{\frac{3}{5}}$$

Divide 2.44 by \frac{3}{5} by multiplying 2.44 by the reciprocal of \frac{3}{5}.

$$t^{2}+\frac{61}{15}t=\frac{2.74}{\frac{3}{5}}$$

Divide 2.74 by \frac{3}{5} by multiplying 2.74 by the reciprocal of \frac{3}{5}.

$$t^{2}+\frac{61}{15}t=\frac{137}{30}$$

Divide \frac{61}{15}, the coefficient of the x term, by 2 to get \frac{61}{30}. Then add the square of \frac{61}{30} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{61}{15}t+\left(\frac{61}{30}\right)^{2}=\frac{137}{30}+\left(\frac{61}{30}\right)^{2}$$

Square \frac{61}{30} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{61}{15}t+\frac{3721}{900}=\frac{137}{30}+\frac{3721}{900}$$

Add \frac{137}{30} to \frac{3721}{900} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}+\frac{61}{15}t+\frac{3721}{900}=\frac{7831}{900}$$

Factor t^{2}+\frac{61}{15}t+\frac{3721}{900}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{61}{30}\right)^{2}=\frac{7831}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{61}{30}\right)^{2}}=\sqrt{\frac{7831}{900}}$$

Simplify.

$$t+\frac{61}{30}=\frac{\sqrt{7831}}{30} t+\frac{61}{30}=-\frac{\sqrt{7831}}{30}$$

Subtract \frac{61}{30} from both sides of the equation.

$$t=\frac{\sqrt{7831}-61}{30} t=\frac{-\sqrt{7831}-61}{30}$$