To solve this equation, we first need to simplify both sides of the equation.
Starting with the left side:
3x/(2x+5) - 28 - 53x/(4x^2-25)= 3x/(2x+5) - 28 - 53x/[(2x+5)(2x-5)] (Difference of squares: 4x^2 - 25 = (2x+5)(2x-5))= [3x(2x-5) - 28(2x+5) - 53x]/[(2x+5)(2x-5)] (Common denominator)= (6x^2 - 15x - 56x - 140 - 53x)/[(2x+5)(2x-5)] (Distribute)= (6x^2 - 124x - 140)/[(2x+5)(2x-5)] (Combine like terms)
Now moving on to the right side:
4x/(2x-5)= 2(2x)/(2x-5)= 4x/(2x-5)
Setting the two sides equal to each other:
(6x^2 - 124x - 140)/[(2x+5)(2x-5)] = 4x/(2x-5)
Now, we can cross multiply to get rid of the denominators:
(6x^2 - 124x - 140) = 4x(2x+5)6x^2 - 124x - 140 = 8x^2 + 20x
Rearranging the equation:
2x^2 + 144x + 140 = 0
This is now a quadratic equation that can be solved using factoring, completing the square, or the quadratic formula.
To solve this equation, we first need to simplify both sides of the equation.
Starting with the left side:
3x/(2x+5) - 28 - 53x/(4x^2-25)
= 3x/(2x+5) - 28 - 53x/[(2x+5)(2x-5)] (Difference of squares: 4x^2 - 25 = (2x+5)(2x-5))
= [3x(2x-5) - 28(2x+5) - 53x]/[(2x+5)(2x-5)] (Common denominator)
= (6x^2 - 15x - 56x - 140 - 53x)/[(2x+5)(2x-5)] (Distribute)
= (6x^2 - 124x - 140)/[(2x+5)(2x-5)] (Combine like terms)
Now moving on to the right side:
4x/(2x-5)
= 2(2x)/(2x-5)
= 4x/(2x-5)
Setting the two sides equal to each other:
(6x^2 - 124x - 140)/[(2x+5)(2x-5)] = 4x/(2x-5)
Now, we can cross multiply to get rid of the denominators:
(6x^2 - 124x - 140) = 4x(2x+5)
6x^2 - 124x - 140 = 8x^2 + 20x
Rearranging the equation:
2x^2 + 144x + 140 = 0
This is now a quadratic equation that can be solved using factoring, completing the square, or the quadratic formula.