Solve b(t)=10^5+10^4t-10^3t^2

$$b ( t ) = 10 ^ { 5 } + 10 ^ { 4 } t - 10 ^ { 3 } t ^ { 2 }$$

Evaluate

$$100000+10000t-1000t^{2}$$

Solution Steps

Calculate 10 to the power of 5 and get 100000.

$$100000+10^{4}t-10^{3}t^{2}$$

Calculate 10 to the power of 4 and get 10000.

$$100000+10000t-10^{3}t^{2}$$

Calculate 10 to the power of 3 and get 1000.

$$100000+10000t-1000t^{2}$$

Factor

$$-1000\left(t-\left(5-5\sqrt{5}\right)\right)\left(t-\left(5\sqrt{5}+5\right)\right)$$

Steps Using the Quadratic Formula

Calculate 10 to the power of 5 and get 100000.

$$factor(100000+10^{4}t-10^{3}t^{2})$$

Calculate 10 to the power of 4 and get 10000.

$$factor(100000+10000t-10^{3}t^{2})$$

Calculate 10 to the power of 3 and get 1000.

$$factor(100000+10000t-1000t^{2})$$

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.

$$-1000t^{2}+10000t+100000=0$$

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.

$$t=\frac{-10000±\sqrt{10000^{2}-4\left(-1000\right)\times 100000}}{2\left(-1000\right)}$$

Square 10000.

$$t=\frac{-10000±\sqrt{100000000-4\left(-1000\right)\times 100000}}{2\left(-1000\right)}$$

Multiply -4 times -1000.

$$t=\frac{-10000±\sqrt{100000000+4000\times 100000}}{2\left(-1000\right)}$$

Multiply 4000 times 100000.

$$t=\frac{-10000±\sqrt{100000000+400000000}}{2\left(-1000\right)}$$

Add 100000000 to 400000000.

$$t=\frac{-10000±\sqrt{500000000}}{2\left(-1000\right)}$$

Take the square root of 500000000.

$$t=\frac{-10000±10000\sqrt{5}}{2\left(-1000\right)}$$

Multiply 2 times -1000.

$$t=\frac{-10000±10000\sqrt{5}}{-2000}$$

Now solve the equation t=\frac{-10000±10000\sqrt{5}}{-2000} when ± is plus. Add -10000 to 10000\sqrt{5}.

$$t=\frac{10000\sqrt{5}-10000}{-2000}$$

Divide -10000+10000\sqrt{5} by -2000.

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

Now solve the equation t=\frac{-10000±10000\sqrt{5}}{-2000} when ± is minus. Subtract 10000\sqrt{5} from -10000.

$$t=\frac{-10000\sqrt{5}-10000}{-2000}$$

Divide -10000-10000\sqrt{5} by -2000.

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

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

$$-1000t^{2}+10000t+100000=-1000\left(t-\left(5-5\sqrt{5}\right)\right)\left(t-\left(5\sqrt{5}+5\right)\right)$$