solve the quadratic equation. 1/2x-3+1/x-5=10/9. ,x is not equal to 3/2,5
Answers
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Given,
1/(2x-3) + 1/(x-5) = 10/9
To find,
The value of x.
Solution,
The values of x after solving the given equation will be 6 and 37/20.
We can easily solve this problem by following the given steps.
According to the question,
1/(2x-3) + 1/(x-5) = 10/9
Taking the LCM of the denominators and adding the two fractions,
(x-5) + (2x-3)/(2x-3) (x-5) = 10/9
3x-8/(2x² -13x + 15) = 10/9 [ Using the identity (x+a)(x+b) = x² + (a+b)x + ab]
Using the cross multiplication method,
9(3x-8) = 10(2x²-13x+15)
27x - 72 = 20x² - 130x + 150 [ Multiplying the numbers into the brackets]
20x² - 130x + 150 = 27x-72
20x² - 130x - 27x + 150 + 72 = 0 ( Moving 27x and 72 from the right-hand side to the left-hand side results in the change of their sign from plus to minus and minus to plus.)
20x² - 157x + 222 = 0
Factorising this equation by splitting the middle term into two terms such that their multiplication will be 20x² × 222 and their addition will give -157x,
20x² - 120x - 37x + 222 = 0
20x (x-6) - 37(x-6) = 0 [ Taking 20x common from the first two terms and -37 from the last two terms]
(x-6) (20x-37) = 0
Equating the brackets to zero,
x-6 = 0, 20x-37 = 0
x = 6, x = 37/20
Hence, the values of x after solving the given equation are 6 and 37/20.