Solve 159.5=100(1+3.908*10^-3t-6-5.802*10^-1t^2)

$$159.5 = 100 ( 1 + 3.908 \times 10 ^ { - 3 } t - 6 - 5.802 \times 10 ^ { - 1 } t ^ { 2 } )$$

Solve for t

$$t=\frac{977+i\sqrt{956603795471}}{290100}\approx 0.003367804+3.371462403i$$

$$t=\frac{-i\sqrt{956603795471}+977}{290100}\approx 0.003367804-3.371462403i$$

Steps Using the Quadratic Formula

Divide both sides by 100.

$$\frac{159.5}{100}=1+3.908\times 10^{-3}t-6-5.802\times 10^{-1}t^{2}$$

Expand \frac{159.5}{100} by multiplying both numerator and the denominator by 10.

$$\frac{1595}{1000}=1+3.908\times 10^{-3}t-6-5.802\times 10^{-1}t^{2}$$

Reduce the fraction \frac{1595}{1000} to lowest terms by extracting and canceling out 5.

$$\frac{319}{200}=1+3.908\times 10^{-3}t-6-5.802\times 10^{-1}t^{2}$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\frac{319}{200}=1+3.908\times \frac{1}{1000}t-6-5.802\times 10^{-1}t^{2}$$

Multiply 3.908 and \frac{1}{1000} to get \frac{977}{250000}.

$$\frac{319}{200}=1+\frac{977}{250000}t-6-5.802\times 10^{-1}t^{2}$$

Subtract 6 from 1 to get -5.

$$\frac{319}{200}=-5+\frac{977}{250000}t-5.802\times 10^{-1}t^{2}$$

Calculate 10 to the power of -1 and get \frac{1}{10}.

$$\frac{319}{200}=-5+\frac{977}{250000}t-5.802\times \frac{1}{10}t^{2}$$

Multiply 5.802 and \frac{1}{10} to get \frac{2901}{5000}.

$$\frac{319}{200}=-5+\frac{977}{250000}t-\frac{2901}{5000}t^{2}$$

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

$$-5+\frac{977}{250000}t-\frac{2901}{5000}t^{2}=\frac{319}{200}$$

Subtract \frac{319}{200} from both sides.

$$-5+\frac{977}{250000}t-\frac{2901}{5000}t^{2}-\frac{319}{200}=0$$

Subtract \frac{319}{200} from -5 to get -\frac{1319}{200}.

$$-\frac{1319}{200}+\frac{977}{250000}t-\frac{2901}{5000}t^{2}=0$$

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.

$$-\frac{2901}{5000}t^{2}+\frac{977}{250000}t-\frac{1319}{200}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{2901}{5000} for a, \frac{977}{250000} for b, and -\frac{1319}{200} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\frac{977}{250000}±\sqrt{\left(\frac{977}{250000}\right)^{2}-4\left(-\frac{2901}{5000}\right)\left(-\frac{1319}{200}\right)}}{2\left(-\frac{2901}{5000}\right)}$$

Square \frac{977}{250000} by squaring both the numerator and the denominator of the fraction.

$$t=\frac{-\frac{977}{250000}±\sqrt{\frac{954529}{62500000000}-4\left(-\frac{2901}{5000}\right)\left(-\frac{1319}{200}\right)}}{2\left(-\frac{2901}{5000}\right)}$$

Multiply -4 times -\frac{2901}{5000}.

$$t=\frac{-\frac{977}{250000}±\sqrt{\frac{954529}{62500000000}+\frac{2901}{1250}\left(-\frac{1319}{200}\right)}}{2\left(-\frac{2901}{5000}\right)}$$

Multiply \frac{2901}{1250} times -\frac{1319}{200} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\frac{977}{250000}±\sqrt{\frac{954529}{62500000000}-\frac{3826419}{250000}}}{2\left(-\frac{2901}{5000}\right)}$$

Add \frac{954529}{62500000000} to -\frac{3826419}{250000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t=\frac{-\frac{977}{250000}±\sqrt{-\frac{956603795471}{62500000000}}}{2\left(-\frac{2901}{5000}\right)}$$

Take the square root of -\frac{956603795471}{62500000000}.

$$t=\frac{-\frac{977}{250000}±\frac{\sqrt{956603795471}i}{250000}}{2\left(-\frac{2901}{5000}\right)}$$

Multiply 2 times -\frac{2901}{5000}.

$$t=\frac{-\frac{977}{250000}±\frac{\sqrt{956603795471}i}{250000}}{-\frac{2901}{2500}}$$

Now solve the equation t=\frac{-\frac{977}{250000}±\frac{\sqrt{956603795471}i}{250000}}{-\frac{2901}{2500}} when ± is plus. Add -\frac{977}{250000} to \frac{i\sqrt{956603795471}}{250000}.

$$t=\frac{-977+\sqrt{956603795471}i}{-\frac{2901}{2500}\times 250000}$$

Divide \frac{-977+i\sqrt{956603795471}}{250000} by -\frac{2901}{2500} by multiplying \frac{-977+i\sqrt{956603795471}}{250000} by the reciprocal of -\frac{2901}{2500}.

$$t=\frac{-\sqrt{956603795471}i+977}{290100}$$

Now solve the equation t=\frac{-\frac{977}{250000}±\frac{\sqrt{956603795471}i}{250000}}{-\frac{2901}{2500}} when ± is minus. Subtract \frac{i\sqrt{956603795471}}{250000} from -\frac{977}{250000}.

$$t=\frac{-\sqrt{956603795471}i-977}{-\frac{2901}{2500}\times 250000}$$

Divide \frac{-977-i\sqrt{956603795471}}{250000} by -\frac{2901}{2500} by multiplying \frac{-977-i\sqrt{956603795471}}{250000} by the reciprocal of -\frac{2901}{2500}.

$$t=\frac{977+\sqrt{956603795471}i}{290100}$$

The equation is now solved.

$$t=\frac{-\sqrt{956603795471}i+977}{290100} t=\frac{977+\sqrt{956603795471}i}{290100}$$

Steps for Completing the Square

Divide both sides by 100.

$$\frac{159.5}{100}=1+3.908\times 10^{-3}t-6-5.802\times 10^{-1}t^{2}$$

Expand \frac{159.5}{100} by multiplying both numerator and the denominator by 10.

$$\frac{1595}{1000}=1+3.908\times 10^{-3}t-6-5.802\times 10^{-1}t^{2}$$

Reduce the fraction \frac{1595}{1000} to lowest terms by extracting and canceling out 5.

$$\frac{319}{200}=1+3.908\times 10^{-3}t-6-5.802\times 10^{-1}t^{2}$$

Calculate 10 to the power of -3 and get \frac{1}{1000}.

$$\frac{319}{200}=1+3.908\times \frac{1}{1000}t-6-5.802\times 10^{-1}t^{2}$$

Multiply 3.908 and \frac{1}{1000} to get \frac{977}{250000}.

$$\frac{319}{200}=1+\frac{977}{250000}t-6-5.802\times 10^{-1}t^{2}$$

Subtract 6 from 1 to get -5.

$$\frac{319}{200}=-5+\frac{977}{250000}t-5.802\times 10^{-1}t^{2}$$

Calculate 10 to the power of -1 and get \frac{1}{10}.

$$\frac{319}{200}=-5+\frac{977}{250000}t-5.802\times \frac{1}{10}t^{2}$$

Multiply 5.802 and \frac{1}{10} to get \frac{2901}{5000}.

$$\frac{319}{200}=-5+\frac{977}{250000}t-\frac{2901}{5000}t^{2}$$

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

$$-5+\frac{977}{250000}t-\frac{2901}{5000}t^{2}=\frac{319}{200}$$

Add 5 to both sides.

$$\frac{977}{250000}t-\frac{2901}{5000}t^{2}=\frac{319}{200}+5$$

Add \frac{319}{200} and 5 to get \frac{1319}{200}.

$$\frac{977}{250000}t-\frac{2901}{5000}t^{2}=\frac{1319}{200}$$

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.

$$-\frac{2901}{5000}t^{2}+\frac{977}{250000}t=\frac{1319}{200}$$

Divide both sides of the equation by -\frac{2901}{5000}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{2901}{5000}t^{2}+\frac{977}{250000}t}{-\frac{2901}{5000}}=\frac{\frac{1319}{200}}{-\frac{2901}{5000}}$$

Dividing by -\frac{2901}{5000} undoes the multiplication by -\frac{2901}{5000}.

$$t^{2}+\frac{\frac{977}{250000}}{-\frac{2901}{5000}}t=\frac{\frac{1319}{200}}{-\frac{2901}{5000}}$$

Divide \frac{977}{250000} by -\frac{2901}{5000} by multiplying \frac{977}{250000} by the reciprocal of -\frac{2901}{5000}.

$$t^{2}-\frac{977}{145050}t=\frac{\frac{1319}{200}}{-\frac{2901}{5000}}$$

Divide \frac{1319}{200} by -\frac{2901}{5000} by multiplying \frac{1319}{200} by the reciprocal of -\frac{2901}{5000}.

$$t^{2}-\frac{977}{145050}t=-\frac{32975}{2901}$$

Divide -\frac{977}{145050}, the coefficient of the x term, by 2 to get -\frac{977}{290100}. Then add the square of -\frac{977}{290100} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$t^{2}-\frac{977}{145050}t+\left(-\frac{977}{290100}\right)^{2}=-\frac{32975}{2901}+\left(-\frac{977}{290100}\right)^{2}$$

Square -\frac{977}{290100} by squaring both the numerator and the denominator of the fraction.

$$t^{2}-\frac{977}{145050}t+\frac{954529}{84158010000}=-\frac{32975}{2901}+\frac{954529}{84158010000}$$

Add -\frac{32975}{2901} to \frac{954529}{84158010000} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$t^{2}-\frac{977}{145050}t+\frac{954529}{84158010000}=-\frac{956603795471}{84158010000}$$

Factor t^{2}-\frac{977}{145050}t+\frac{954529}{84158010000}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{977}{290100}\right)^{2}=-\frac{956603795471}{84158010000}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(t-\frac{977}{290100}\right)^{2}}=\sqrt{-\frac{956603795471}{84158010000}}$$

Simplify.

$$t-\frac{977}{290100}=\frac{\sqrt{956603795471}i}{290100} t-\frac{977}{290100}=-\frac{\sqrt{956603795471}i}{290100}$$

Add \frac{977}{290100} to both sides of the equation.

$$t=\frac{977+\sqrt{956603795471}i}{290100} t=\frac{-\sqrt{956603795471}i+977}{290100}$$