Solve 100=2t+5t^2

$$100 = 2 t + 5 t ^ { 2 }$$

Solve for t

$$t = \frac{\sqrt{501} - 1}{5} \approx 4.276605857$$

$$t=\frac{-\sqrt{501}-1}{5}\approx -4.676605857$$

Steps Using the Quadratic Formula

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

$$2t+5t^{2}=100$$

Subtract 100 from both sides.

$$2t+5t^{2}-100=0$$

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.

$$5t^{2}+2t-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 2 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-2±\sqrt{2^{2}-4\times 5\left(-100\right)}}{2\times 5}$$

Square 2.

$$t=\frac{-2±\sqrt{4-4\times 5\left(-100\right)}}{2\times 5}$$

Multiply -4 times 5.

$$t=\frac{-2±\sqrt{4-20\left(-100\right)}}{2\times 5}$$

Multiply -20 times -100.

$$t=\frac{-2±\sqrt{4+2000}}{2\times 5}$$

Add 4 to 2000.

$$t=\frac{-2±\sqrt{2004}}{2\times 5}$$

Take the square root of 2004.

$$t=\frac{-2±2\sqrt{501}}{2\times 5}$$

Multiply 2 times 5.

$$t=\frac{-2±2\sqrt{501}}{10}$$

Now solve the equation t=\frac{-2±2\sqrt{501}}{10} when ± is plus. Add -2 to 2\sqrt{501}.

$$t=\frac{2\sqrt{501}-2}{10}$$

Divide -2+2\sqrt{501} by 10.

$$t=\frac{\sqrt{501}-1}{5}$$

Now solve the equation t=\frac{-2±2\sqrt{501}}{10} when ± is minus. Subtract 2\sqrt{501} from -2.

$$t=\frac{-2\sqrt{501}-2}{10}$$

Divide -2-2\sqrt{501} by 10.

$$t=\frac{-\sqrt{501}-1}{5}$$

The equation is now solved.

$$t=\frac{\sqrt{501}-1}{5} t=\frac{-\sqrt{501}-1}{5}$$

Steps for Completing the Square

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

$$2t+5t^{2}=100$$

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.

$$5t^{2}+2t=100$$

Divide both sides by 5.

$$\frac{5t^{2}+2t}{5}=\frac{100}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$t^{2}+\frac{2}{5}t=\frac{100}{5}$$

Divide 100 by 5.

$$t^{2}+\frac{2}{5}t=20$$

Divide \frac{2}{5}, the coefficient of the x term, by 2 to get \frac{1}{5}. Then add the square of \frac{1}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}+\frac{2}{5}t+\left(\frac{1}{5}\right)^{2}=20+\left(\frac{1}{5}\right)^{2}$$

Square \frac{1}{5} by squaring both the numerator and the denominator of the fraction.

$$t^{2}+\frac{2}{5}t+\frac{1}{25}=20+\frac{1}{25}$$

Add 20 to \frac{1}{25}.

$$t^{2}+\frac{2}{5}t+\frac{1}{25}=\frac{501}{25}$$

Factor t^{2}+\frac{2}{5}t+\frac{1}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{1}{5}\right)^{2}=\frac{501}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t+\frac{1}{5}\right)^{2}}=\sqrt{\frac{501}{25}}$$

Simplify.

$$t+\frac{1}{5}=\frac{\sqrt{501}}{5} t+\frac{1}{5}=-\frac{\sqrt{501}}{5}$$

Subtract \frac{1}{5} from both sides of the equation.

$$t=\frac{\sqrt{501}-1}{5} t=\frac{-\sqrt{501}-1}{5}$$