Solve h(t)=100+40t-4t^2

$$h ( t ) = 100 + 40 t - 4 t ^ { 2 }$$

Factor

$$-4\left(t-\left(5-5\sqrt{2}\right)\right)\left(t-\left(5\sqrt{2}+5\right)\right)$$

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.

$$-4t^{2}+40t+100=0$$

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.

$$t=\frac{-40±\sqrt{40^{2}-4\left(-4\right)\times 100}}{2\left(-4\right)}$$

Square 40.

$$t=\frac{-40±\sqrt{1600-4\left(-4\right)\times 100}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$t=\frac{-40±\sqrt{1600+16\times 100}}{2\left(-4\right)}$$

Multiply 16 times 100.

$$t=\frac{-40±\sqrt{1600+1600}}{2\left(-4\right)}$$

Add 1600 to 1600.

$$t=\frac{-40±\sqrt{3200}}{2\left(-4\right)}$$

Take the square root of 3200.

$$t=\frac{-40±40\sqrt{2}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$t=\frac{-40±40\sqrt{2}}{-8}$$

Now solve the equation t=\frac{-40±40\sqrt{2}}{-8} when ± is plus. Add -40 to 40\sqrt{2}.

$$t=\frac{40\sqrt{2}-40}{-8}$$

Divide -40+40\sqrt{2} by -8.

$$t=5-5\sqrt{2}$$

Now solve the equation t=\frac{-40±40\sqrt{2}}{-8} when ± is minus. Subtract 40\sqrt{2} from -40.

$$t=\frac{-40\sqrt{2}-40}{-8}$$

Divide -40-40\sqrt{2} by -8.

$$t=5\sqrt{2}+5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5-5\sqrt{2} for x_{1} and 5+5\sqrt{2} for x_{2}.

$$-4t^{2}+40t+100=-4\left(t-\left(5-5\sqrt{2}\right)\right)\left(t-\left(5\sqrt{2}+5\right)\right)$$

Evaluate

$$100+40t-4t^{2}$$