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Popular >

((2t+5)(3t-1))/3+(t^2+5)/2 =(7t-5)/6

Solution

\frac{( 2 t + 5 ) ( 3 t - 1 )}{3} + \frac{t ^{2} + 5}{2} = \frac{7 t - 5}{6}

Solution

t = - \frac{19}{30} + i \frac{239}{30}, t = - \frac{19}{30} - i \frac{239}{30}

Solution steps

\frac{( 2 t + 5 ) ( 3 t - 1 )}{3} + \frac{t ^{2} + 5}{2} = \frac{7 t - 5}{6}

Find Least Common Multiplier of

3, 2, 6 : 6

Multiply by LCM=

6

\frac{( 2 t + 5 ) ( 3 t - 1 )}{3} \cdot 6 + \frac{t ^{2} + 5}{2} \cdot 6 = \frac{7 t - 5}{6} \cdot 6

Simplify

2 (2 t + 5) (3 t - 1) + 3 (t^{2} + 5) = 7 t - 5

Expand

2 (2 t + 5) (3 t - 1) + 3 (t^{2} + 5) : 15 t^{2} + 26 t + 5

15 t^{2} + 26 t + 5 = 7 t - 5

Move

5

to the left side

15 t^{2} + 26 t + 10 = 7 t

Move

7 t

to the left side

15 t^{2} + 19 t + 10 = 0

Solve with the quadratic formula

t_{1, 2} = \frac{- 19 \pm 1 9 ^{2} - 4 \cdot 15 \cdot 10}{2 \cdot 15}

Simplify

1 9^{2} - 4 \cdot 15 \cdot 10 : 239 i

t_{1, 2} = \frac{- 19 \pm 239 i}{2 \cdot 15}

Separate the solutions

t_{1} = \frac{- 19 + 239 i}{2 \cdot 15}, t_{2} = \frac{- 19 - 239 i}{2 \cdot 15}

t = \frac{- 19 + 239 i}{2 \cdot 15} : - \frac{19}{30} + i \frac{239}{30}

t = \frac{- 19 - 239 i}{2 \cdot 15} : - \frac{19}{30} - i \frac{239}{30}

The solutions to the quadratic equation are:

t = - \frac{19}{30} + i \frac{239}{30}, t = - \frac{19}{30} - i \frac{239}{30}

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