Solve 255=(-39,2)t+(0,5*9,8)t^2

$$255 = ( - 39,2 ) t + ( 0,5 \cdot 9,8 ) t ^ { 2 }$$

Solve for t

$$t=\frac{\sqrt{3334}}{7}+4\approx 12.248685733$$

$$t=-\frac{\sqrt{3334}}{7}+4\approx -4.248685733$$

Steps Using the Quadratic Formula

Multiply 0,5 and 9,8 to get 4,9.

$$255=-39,2t+4,9t^{2}$$

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

$$-39,2t+4,9t^{2}=255$$

Subtract 255 from both sides.

$$-39,2t+4,9t^{2}-255=0$$

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.

$$4,9t^{2}-39,2t-255=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4,9 for a, -39,2 for b, and -255 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-39,2\right)±\sqrt{\left(-39,2\right)^{2}-4\times 4,9\left(-255\right)}}{2\times 4,9}$$

Square -39,2 by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\left(-39,2\right)±\sqrt{1536,64-4\times 4,9\left(-255\right)}}{2\times 4,9}$$

Multiply -4 times 4,9.

$$t=\frac{-\left(-39,2\right)±\sqrt{1536,64-19,6\left(-255\right)}}{2\times 4,9}$$

Multiply -19,6 times -255.

$$t=\frac{-\left(-39,2\right)±\sqrt{1536,64+4998}}{2\times 4,9}$$

Add 1536,64 to 4998.

$$t=\frac{-\left(-39,2\right)±\sqrt{6534,64}}{2\times 4,9}$$

Take the square root of 6534,64.

$$t=\frac{-\left(-39,2\right)±\frac{7\sqrt{3334}}{5}}{2\times 4,9}$$

The opposite of -39,2 is 39,2.

$$t=\frac{39,2±\frac{7\sqrt{3334}}{5}}{2\times 4,9}$$

Multiply 2 times 4,9.

$$t=\frac{39,2±\frac{7\sqrt{3334}}{5}}{9,8}$$

Now solve the equation t=\frac{39,2±\frac{7\sqrt{3334}}{5}}{9,8} when ± is plus. Add 39,2 to \frac{7\sqrt{3334}}{5}.

$$t=\frac{7\sqrt{3334}+196}{5\times 9,8}$$

Divide \frac{196+7\sqrt{3334}}{5} by 9,8 by multiplying \frac{196+7\sqrt{3334}}{5} by the reciprocal of 9,8.

$$t=\frac{\sqrt{3334}}{7}+4$$

Now solve the equation t=\frac{39,2±\frac{7\sqrt{3334}}{5}}{9,8} when ± is minus. Subtract \frac{7\sqrt{3334}}{5} from 39,2.

$$t=\frac{196-7\sqrt{3334}}{5\times 9,8}$$

Divide \frac{196-7\sqrt{3334}}{5} by 9,8 by multiplying \frac{196-7\sqrt{3334}}{5} by the reciprocal of 9,8.

$$t=-\frac{\sqrt{3334}}{7}+4$$

The equation is now solved.

$$t=\frac{\sqrt{3334}}{7}+4 t=-\frac{\sqrt{3334}}{7}+4$$

Steps for Completing the Square

Multiply 0,5 and 9,8 to get 4,9.

$$255=-39,2t+4,9t^{2}$$

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

$$-39,2t+4,9t^{2}=255$$

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.

$$4,9t^{2}-39,2t=255$$

Divide both sides of the equation by 4,9, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{4,9t^{2}-39,2t}{4,9}=\frac{255}{4,9}$$

Dividing by 4,9 undoes the multiplication by 4,9.

$$t^{2}+\left(-\frac{39,2}{4,9}\right)t=\frac{255}{4,9}$$

Divide -39,2 by 4,9 by multiplying -39,2 by the reciprocal of 4,9.

$$t^{2}-8t=\frac{255}{4,9}$$

Divide 255 by 4,9 by multiplying 255 by the reciprocal of 4,9.

$$t^{2}-8t=\frac{2550}{49}$$

Divide -8, the coefficient of the x term, by 2 to get -4. Then add the square of -4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-8t+\left(-4\right)^{2}=\frac{2550}{49}+\left(-4\right)^{2}$$

Square -4.

$$t^{2}-8t+16=\frac{2550}{49}+16$$

Add \frac{2550}{49} to 16.

$$t^{2}-8t+16=\frac{3334}{49}$$

Factor t^{2}-8t+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-4\right)^{2}=\frac{3334}{49}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-4\right)^{2}}=\sqrt{\frac{3334}{49}}$$

Simplify.

$$t-4=\frac{\sqrt{3334}}{7} t-4=-\frac{\sqrt{3334}}{7}$$

Add 4 to both sides of the equation.

$$t=\frac{\sqrt{3334}}{7}+4 t=-\frac{\sqrt{3334}}{7}+4$$