Solve 3000+800t-12t^2

$$3000 + 800 t - 12 t ^ { 2 }$$

Factor

$$-12\left(t-\frac{100-35\sqrt{10}}{3}\right)\left(t-\frac{35\sqrt{10}+100}{3}\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-12t^{2}+800t+3000=0$$

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.

$$t=\frac{-800±\sqrt{800^{2}-4\left(-12\right)\times 3000}}{2\left(-12\right)}$$

Square 800.

$$t=\frac{-800±\sqrt{640000-4\left(-12\right)\times 3000}}{2\left(-12\right)}$$

Multiply -4 times -12.

$$t=\frac{-800±\sqrt{640000+48\times 3000}}{2\left(-12\right)}$$

Multiply 48 times 3000.

$$t=\frac{-800±\sqrt{640000+144000}}{2\left(-12\right)}$$

Add 640000 to 144000.

$$t=\frac{-800±\sqrt{784000}}{2\left(-12\right)}$$

Take the square root of 784000.

$$t=\frac{-800±280\sqrt{10}}{2\left(-12\right)}$$

Multiply 2 times -12.

$$t=\frac{-800±280\sqrt{10}}{-24}$$

Now solve the equation t=\frac{-800±280\sqrt{10}}{-24} when ± is plus. Add -800 to 280\sqrt{10}.

$$t=\frac{280\sqrt{10}-800}{-24}$$

Divide -800+280\sqrt{10} by -24.

$$t=\frac{100-35\sqrt{10}}{3}$$

Now solve the equation t=\frac{-800±280\sqrt{10}}{-24} when ± is minus. Subtract 280\sqrt{10} from -800.

$$t=\frac{-280\sqrt{10}-800}{-24}$$

Divide -800-280\sqrt{10} by -24.

$$t=\frac{35\sqrt{10}+100}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{100-35\sqrt{10}}{3} for x_{1} and \frac{100+35\sqrt{10}}{3} for x_{2}.

$$-12t^{2}+800t+3000=-12\left(t-\frac{100-35\sqrt{10}}{3}\right)\left(t-\frac{35\sqrt{10}+100}{3}\right)$$

Evaluate

$$3000+800t-12t^{2}$$