Question

Solve the equation and simplify your answer: ((x)/(x+5)+(3x)/(x²+7x+10)=(2)/(x+2)

Answer 1

Therefore the answer is
`\[ x = -(5)/(2) \]`

Step-by-step explanation

The given equation is
`\((x)/(x+5) + (3x)/(x^2+7x+10) = (2)/(x+2)\).`To solve this equation, we'll first find a common denominator. The denominators in this case are
`\((x+5)\), \((x+2)\), and \((x^2+7x+10)\).` The common denominator is
`\((x+5)(x+2)(x+5)\).` Now, multiply each term by the missing factors to clear the fractions.

`\[ x(x+2)(x+2) + 3x(x+5) = 2(x+5)(x+2) \]`

Next, simplify and combine like terms:

`\[ x(x^2+4x+4) + 3x^2 + 15x = 2(x^2+7x+10) \]`

Expand and collect like terms:

`\[ x^3 + 4x^2 + 4x + 3x^2 + 15x = 2x^2 + 14x + 20 \]`

Combine like terms again:

`\[ x^3 + 7x^2 + 19x - 2x^2 - 14x - 20 = 0 \]`

Finally, simplify the equation:

`\[ x^3 + 5x^2 + 5x - 20 = 0 \]`

Now, factor the cubic equation or use numerical methods to find the solutions. The solution is
`\( x = -(5)/(2) \).`

Therefore, the final answer to the given equation is
`\( x = -(5)/(2) \).`