Solve (10/t)-(100/(t-180)) = 10

Question

Solve the equation

t_{1} = \frac{171 - 3 3169}{2}, t_{2} = \frac{171 + 3 3169}{2}

Alternative Form

t_{1} \approx 1.059192, t_{2} \approx 169.940808

Evaluate

\frac{10}{t} - (\frac{100}{t - 180}) = 10

Find the domain

More Steps

\frac{10}{t} - (\frac{100}{t - 180}) = 10, t \in (- \infty, 0) \cup (0, 180) \cup (180, + \infty)

Remove the unnecessary parentheses

\frac{10}{t} - \frac{100}{t - 180} = 10

Multiply both sides of the equation by LCD

(\frac{10}{t} - \frac{100}{t - 180}) t (t - 180) = 10 t (t - 180)

Simplify the equation

More Steps

- 90 t - 1800 = 10 t (t - 180)

Simplify the equation

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- 90 t - 1800 = 10 t^{2} - 1800 t

Move the expression to the left side

- 90 t - 1800 - (10 t^{2} - 1800 t) = 0

Subtract the terms

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1710 t - 1800 - 10 t^{2} = 0

Rewrite in standard form

- 10 t^{2} + 1710 t - 1800 = 0

Multiply both sides

10 t^{2} - 1710 t + 1800 = 0

Substitute a= 10,b= - 1710 and c= 1800 into the quadratic formula t = \frac{- b \pm b ^{2} - 4 a c}{2 a}

t = \frac{1710 \pm ( - 1710 ) ^{2} - 4 \times 10 \times 1800}{2 \times 10}

Simplify the expression

t = \frac{1710 \pm ( - 1710 ) ^{2} - 4 \times 10 \times 1800}{20}

Simplify the expression

More Steps

t = \frac{1710 \pm 171 0 ^{2} - 72000}{20}

Simplify the radical expression

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t = \frac{1710 \pm 30 3169}{20}

Separate the equation into 2 possible cases

t = \frac{1710 + 30 3169}{20} t = \frac{1710 - 30 3169}{20}

Simplify the expression

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t = \frac{171 + 3 3169}{2} t = \frac{1710 - 30 3169}{20}

Simplify the expression

More Steps

t = \frac{171 + 3 3169}{2} t = \frac{171 - 3 3169}{2}

Check if the solution is in the defined range

t = \frac{171 + 3 3169}{2} t = \frac{171 - 3 3169}{2}, t \in (- \infty, 0) \cup (0, 180) \cup (180, + \infty)

Find the intersection of the solution and the defined range

t = \frac{171 + 3 3169}{2} t = \frac{171 - 3 3169}{2}

Solution

t_{1} = \frac{171 - 3 3169}{2}, t_{2} = \frac{171 + 3 3169}{2}

Alternative Form

t_{1} \approx 1.059192, t_{2} \approx 169.940808

Show Solution