Solve -10460/1373=-39155.7*(1/1373-1/T)

$$\frac { - 10460 } { 1373 } = - 39155.7 \times ( \frac { 1 } { 1373 } - \frac { 1 } { T } )$$

Solve for T

$$T = \frac{391557}{209} = 1873\frac{100}{209} \approx 1873.4784689$$

Steps Using Factoring

Variable T cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1373T, the least common multiple of 1373,T.

$$T\left(-10460\right)=-39155.7\left(\frac{1}{1373}-\frac{1}{T}\right)\times 1373T$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1373 and T is 1373T. Multiply \frac{1}{1373} times \frac{T}{T}. Multiply \frac{1}{T} times \frac{1373}{1373}.

$$T\left(-10460\right)=-39155.7\left(\frac{T}{1373T}-\frac{1373}{1373T}\right)\times 1373T$$

Since \frac{T}{1373T} and \frac{1373}{1373T} have the same denominator, subtract them by subtracting their numerators.

$$T\left(-10460\right)=-39155.7\times \frac{T-1373}{1373T}\times 1373T$$

Multiply -39155.7 and 1373 to get -53760776.1.

$$T\left(-10460\right)=-53760776.1\times \frac{T-1373}{1373T}T$$

Express \frac{T-1373}{1373T}T as a single fraction.

$$T\left(-10460\right)=-53760776.1\times \frac{\left(T-1373\right)T}{1373T}$$

Use the distributive property to multiply T-1373 by T.

$$T\left(-10460\right)=-53760776.1\times \frac{T^{2}-1373T}{1373T}$$

Add 53760776.1\times \frac{T^{2}-1373T}{1373T} to both sides.

$$T\left(-10460\right)+53760776.1\times \frac{T^{2}-1373T}{1373T}=0$$

Variable T cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1373T.

$$T\left(-10460\right)\times 1373T+53760776.1\left(T^{2}-1373T\right)=0$$

Reorder the terms.

$$53760776.1\left(T^{2}-1373T\right)-10460\times 1373TT=0$$

Multiply T and T to get T^{2}.

$$53760776.1\left(T^{2}-1373T\right)-10460\times 1373T^{2}=0$$

Use the distributive property to multiply 53760776.1 by T^{2}-1373T.

$$53760776.1T^{2}-73813545585.3T-10460\times 1373T^{2}=0$$

Multiply -10460 and 1373 to get -14361580.

$$53760776.1T^{2}-73813545585.3T-14361580T^{2}=0$$

Combine 53760776.1T^{2} and -14361580T^{2} to get 39399196.1T^{2}.

$$39399196.1T^{2}-73813545585.3T=0$$

Factor out T.

$$T\left(39399196.1T-73813545585.3\right)=0$$

To find equation solutions, solve T=0 and \frac{393991961T-738135455853}{10}=0.

$$T=0 T=\frac{391557}{209}$$

Variable T cannot be equal to 0.

$$T=\frac{391557}{209}$$

Steps Using the Quadratic Formula

Variable T cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1373T, the least common multiple of 1373,T.

$$T\left(-10460\right)=-39155.7\left(\frac{1}{1373}-\frac{1}{T}\right)\times 1373T$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1373 and T is 1373T. Multiply \frac{1}{1373} times \frac{T}{T}. Multiply \frac{1}{T} times \frac{1373}{1373}.

$$T\left(-10460\right)=-39155.7\left(\frac{T}{1373T}-\frac{1373}{1373T}\right)\times 1373T$$

Since \frac{T}{1373T} and \frac{1373}{1373T} have the same denominator, subtract them by subtracting their numerators.

$$T\left(-10460\right)=-39155.7\times \frac{T-1373}{1373T}\times 1373T$$

Multiply -39155.7 and 1373 to get -53760776.1.

$$T\left(-10460\right)=-53760776.1\times \frac{T-1373}{1373T}T$$

Express \frac{T-1373}{1373T}T as a single fraction.

$$T\left(-10460\right)=-53760776.1\times \frac{\left(T-1373\right)T}{1373T}$$

Use the distributive property to multiply T-1373 by T.

$$T\left(-10460\right)=-53760776.1\times \frac{T^{2}-1373T}{1373T}$$

Add 53760776.1\times \frac{T^{2}-1373T}{1373T} to both sides.

$$T\left(-10460\right)+53760776.1\times \frac{T^{2}-1373T}{1373T}=0$$

Variable T cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1373T.

$$T\left(-10460\right)\times 1373T+53760776.1\left(T^{2}-1373T\right)=0$$

Reorder the terms.

$$53760776.1\left(T^{2}-1373T\right)-10460\times 1373TT=0$$

Multiply T and T to get T^{2}.

$$53760776.1\left(T^{2}-1373T\right)-10460\times 1373T^{2}=0$$

Use the distributive property to multiply 53760776.1 by T^{2}-1373T.

$$53760776.1T^{2}-73813545585.3T-10460\times 1373T^{2}=0$$

Multiply -10460 and 1373 to get -14361580.

$$53760776.1T^{2}-73813545585.3T-14361580T^{2}=0$$

Combine 53760776.1T^{2} and -14361580T^{2} to get 39399196.1T^{2}.

$$39399196.1T^{2}-73813545585.3T=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 39399196.1 for a, -73813545585.3 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$T=\frac{-\left(-73813545585.3\right)±\sqrt{\left(-73813545585.3\right)^{2}}}{2\times 39399196.1}$$

Take the square root of \left(-73813545585.3\right)^{2}.

$$T=\frac{-\left(-73813545585.3\right)±\frac{738135455853}{10}}{2\times 39399196.1}$$

The opposite of -73813545585.3 is 73813545585.3.

$$T=\frac{73813545585.3±\frac{738135455853}{10}}{2\times 39399196.1}$$

Multiply 2 times 39399196.1.

$$T=\frac{73813545585.3±\frac{738135455853}{10}}{78798392.2}$$

Now solve the equation T=\frac{73813545585.3±\frac{738135455853}{10}}{78798392.2} when ± is plus. Add 73813545585.3 to \frac{738135455853}{10} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$T=\frac{\frac{738135455853}{5}}{78798392.2}$$

Divide \frac{738135455853}{5} by 78798392.2 by multiplying \frac{738135455853}{5} by the reciprocal of 78798392.2.

$$T=\frac{391557}{209}$$

Now solve the equation T=\frac{73813545585.3±\frac{738135455853}{10}}{78798392.2} when ± is minus. Subtract \frac{738135455853}{10} from 73813545585.3 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$T=\frac{0}{78798392.2}$$

Divide 0 by 78798392.2 by multiplying 0 by the reciprocal of 78798392.2.

$$T=0$$

The equation is now solved.

$$T=\frac{391557}{209} T=0$$

Variable T cannot be equal to 0.

$$T=\frac{391557}{209}$$

Steps for Completing the Square

Variable T cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1373T, the least common multiple of 1373,T.

$$T\left(-10460\right)=-39155.7\left(\frac{1}{1373}-\frac{1}{T}\right)\times 1373T$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 1373 and T is 1373T. Multiply \frac{1}{1373} times \frac{T}{T}. Multiply \frac{1}{T} times \frac{1373}{1373}.

$$T\left(-10460\right)=-39155.7\left(\frac{T}{1373T}-\frac{1373}{1373T}\right)\times 1373T$$

Since \frac{T}{1373T} and \frac{1373}{1373T} have the same denominator, subtract them by subtracting their numerators.

$$T\left(-10460\right)=-39155.7\times \frac{T-1373}{1373T}\times 1373T$$

Multiply -39155.7 and 1373 to get -53760776.1.

$$T\left(-10460\right)=-53760776.1\times \frac{T-1373}{1373T}T$$

Express \frac{T-1373}{1373T}T as a single fraction.

$$T\left(-10460\right)=-53760776.1\times \frac{\left(T-1373\right)T}{1373T}$$

Use the distributive property to multiply T-1373 by T.

$$T\left(-10460\right)=-53760776.1\times \frac{T^{2}-1373T}{1373T}$$

Add 53760776.1\times \frac{T^{2}-1373T}{1373T} to both sides.

$$T\left(-10460\right)+53760776.1\times \frac{T^{2}-1373T}{1373T}=0$$

Variable T cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 1373T.

$$T\left(-10460\right)\times 1373T+53760776.1\left(T^{2}-1373T\right)=0$$

Reorder the terms.

$$53760776.1\left(T^{2}-1373T\right)-10460\times 1373TT=0$$

Multiply T and T to get T^{2}.

$$53760776.1\left(T^{2}-1373T\right)-10460\times 1373T^{2}=0$$

Use the distributive property to multiply 53760776.1 by T^{2}-1373T.

$$53760776.1T^{2}-73813545585.3T-10460\times 1373T^{2}=0$$

Multiply -10460 and 1373 to get -14361580.

$$53760776.1T^{2}-73813545585.3T-14361580T^{2}=0$$

Combine 53760776.1T^{2} and -14361580T^{2} to get 39399196.1T^{2}.

$$39399196.1T^{2}-73813545585.3T=0$$

Divide both sides of the equation by 39399196.1, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{39399196.1T^{2}-73813545585.3T}{39399196.1}=\frac{0}{39399196.1}$$

Dividing by 39399196.1 undoes the multiplication by 39399196.1.

$$T^{2}+\left(-\frac{73813545585.3}{39399196.1}\right)T=\frac{0}{39399196.1}$$

Divide -73813545585.3 by 39399196.1 by multiplying -73813545585.3 by the reciprocal of 39399196.1.

$$T^{2}-\frac{391557}{209}T=\frac{0}{39399196.1}$$

Divide 0 by 39399196.1 by multiplying 0 by the reciprocal of 39399196.1.

$$T^{2}-\frac{391557}{209}T=0$$

Divide -\frac{391557}{209}, the coefficient of the x term, by 2 to get -\frac{391557}{418}. Then add the square of -\frac{391557}{418} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$T^{2}-\frac{391557}{209}T+\left(-\frac{391557}{418}\right)^{2}=\left(-\frac{391557}{418}\right)^{2}$$

Square -\frac{391557}{418} by squaring both the numerator and the denominator of the fraction.

$$T^{2}-\frac{391557}{209}T+\frac{153316884249}{174724}=\frac{153316884249}{174724}$$

Factor T^{2}-\frac{391557}{209}T+\frac{153316884249}{174724}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(T-\frac{391557}{418}\right)^{2}=\frac{153316884249}{174724}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(T-\frac{391557}{418}\right)^{2}}=\sqrt{\frac{153316884249}{174724}}$$

Simplify.

$$T-\frac{391557}{418}=\frac{391557}{418} T-\frac{391557}{418}=-\frac{391557}{418}$$

Add \frac{391557}{418} to both sides of the equation.

$$T=\frac{391557}{209} T=0$$

Variable T cannot be equal to 0.

$$T=\frac{391557}{209}$$