Solve 301+2t^2=300t

$$301 + 2 t ^ { 2 } = 300 t$$

Solve for t

$$t=\frac{\sqrt{21898}}{2}+75\approx 148.989864171$$

$$t=-\frac{\sqrt{21898}}{2}+75\approx 1.010135829$$

Steps Using the Quadratic Formula

Subtract 300t from both sides.

$$301+2t^{2}-300t=0$$

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.

$$2t^{2}-300t+301=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -300 for b, and 301 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-300\right)±\sqrt{\left(-300\right)^{2}-4\times 2\times 301}}{2\times 2}$$

Square -300.

$$t=\frac{-\left(-300\right)±\sqrt{90000-4\times 2\times 301}}{2\times 2}$$

Multiply -4 times 2.

$$t=\frac{-\left(-300\right)±\sqrt{90000-8\times 301}}{2\times 2}$$

Multiply -8 times 301.

$$t=\frac{-\left(-300\right)±\sqrt{90000-2408}}{2\times 2}$$

Add 90000 to -2408.

$$t=\frac{-\left(-300\right)±\sqrt{87592}}{2\times 2}$$

Take the square root of 87592.

$$t=\frac{-\left(-300\right)±2\sqrt{21898}}{2\times 2}$$

The opposite of -300 is 300.

$$t=\frac{300±2\sqrt{21898}}{2\times 2}$$

Multiply 2 times 2.

$$t=\frac{300±2\sqrt{21898}}{4}$$

Now solve the equation t=\frac{300±2\sqrt{21898}}{4} when ± is plus. Add 300 to 2\sqrt{21898}.

$$t=\frac{2\sqrt{21898}+300}{4}$$

Divide 300+2\sqrt{21898} by 4.

$$t=\frac{\sqrt{21898}}{2}+75$$

Now solve the equation t=\frac{300±2\sqrt{21898}}{4} when ± is minus. Subtract 2\sqrt{21898} from 300.

$$t=\frac{300-2\sqrt{21898}}{4}$$

Divide 300-2\sqrt{21898} by 4.

$$t=-\frac{\sqrt{21898}}{2}+75$$

The equation is now solved.

$$t=\frac{\sqrt{21898}}{2}+75 t=-\frac{\sqrt{21898}}{2}+75$$

Steps for Completing the Square

Subtract 300t from both sides.

$$301+2t^{2}-300t=0$$

Subtract 301 from both sides. Anything subtracted from zero gives its negation.

$$2t^{2}-300t=-301$$

Divide both sides by 2.

$$\frac{2t^{2}-300t}{2}=-\frac{301}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$t^{2}+\left(-\frac{300}{2}\right)t=-\frac{301}{2}$$

Divide -300 by 2.

$$t^{2}-150t=-\frac{301}{2}$$

Divide -150, the coefficient of the x term, by 2 to get -75. Then add the square of -75 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-150t+\left(-75\right)^{2}=-\frac{301}{2}+\left(-75\right)^{2}$$

Square -75.

$$t^{2}-150t+5625=-\frac{301}{2}+5625$$

Add -\frac{301}{2} to 5625.

$$t^{2}-150t+5625=\frac{10949}{2}$$

Factor t^{2}-150t+5625. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-75\right)^{2}=\frac{10949}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-75\right)^{2}}=\sqrt{\frac{10949}{2}}$$

Simplify.

$$t-75=\frac{\sqrt{21898}}{2} t-75=-\frac{\sqrt{21898}}{2}$$

Add 75 to both sides of the equation.

$$t=\frac{\sqrt{21898}}{2}+75 t=-\frac{\sqrt{21898}}{2}+75$$