Question

2x+3 X
———- = ——
X+5 2x-3 Solving rational equations

Answer 1

Answer:

The solutions to the equation
`\( (2x)/(x+5) = (3)/(2x-3) \)` are:

1.
`\( x = (9)/(8) - (√(321))/(8) \)`

2.
`\( x = (9)/(8) + (√(321))/(8) \)`

Explanation:

1. Cross-multiply: Cross-multiplication is a method used to eliminate the fraction from an equation. It involves multiplying both sides of the equation by the denominators of the fractions. In this case, we multiply both sides by
`\((x+5)\) and \((2x-3)\)` to get rid of the fractions.

So, we get:
`\(2x * (2x - 3) = 3 * (x + 5)\)`

2. Expand: Next, we expand both sides of the equation:

`\(4x^2 - 6x = 3x + 15\)`

3. Rearrange: We then rearrange the equation to bring all terms to one side, which gives us a quadratic equation:

`\(4x^2 - 6x - 3x - 15 = 0\)`

Simplifying, we get:

`\(4x^2 - 9x - 15 = 0\)`

4. Solve the quadratic equation: We can solve this quadratic equation using the quadratic formula
`\(x = (-b \pm √(b^2 - 4ac))/(2a)\)`, where
`\(a = 4\), \(b = -9\)`, and
`\(c = -15\)`.

Substituting these values into the formula, we get:

`\(x = (9 \pm √((-9)^2 - 4*4*(-15)))/(2*4)\)`

Simplifying further, we get:

`\(x = (9 \pm √(321))/(8)\)`

So, the solutions to the equation are
`\(x = (9)/(8) - (√(321))/(8)\)` and
`\(x = (9)/(8) + (√(321))/(8)\)`.

Remember, these solutions are valid as long as
`\(x \\eq -5\)` and
`\(x \\eq (3)/(2)\)` to avoid division by zero in the original equation.