Technique explanation
This lesson extends the merger principle to Complex Merger Equations, which have two terms on each side where the total numerator sums match (A + B = C + D).
In a complex merger, pair terms with equal numerators across the equals sign. For example, moving terms in 3/x + 5/(x+1) = 3/(x+2) + 5/(x+3) gives 3(1/x - 1/(x+2)) + 5(1/(x+1) - 1/(x+3)) = 0. This creates a structural cancellation that drops the degree from three to two.
This method simplifies the solving process by reducing the equation to a quadratic. If the numerator sums do not match, the cubic terms will not cancel.
Standard fallback: Combining all rational terms onto one side of the equation, finding a common denominator for the entire expression, and solving the resulting high-degree polynomial equation.
Worked examples
Example 1: Solve 3/x + 5/(x+1) = 3/(x+2) + 5/(x+3)
- Step 1: Pair identical numerators: 3(1/x - 1/(x+2)) + 5(1/(x+1) - 1/(x+3)) = 0.
- Step 2: Simplify: 3(2)/(x(x+2)) + 5(2)/((x+1)(x+3)) = 0 → 6/(x² + 2x) + 10/(x² + 4x + 3) = 0.
- Step 3: 6(x² + 4x + 3) + 10(x² + 2x) = 0 → 16x² + 44x + 18 = 0 → 8x² + 22x + 9 = 0.
- Step 4: (2x + 1)(4x + 9) = 0 → x = -1/2, -9/4.
- Check: Substitute x = -1/2 into the LHS: 3/(-0.5) + 5/(0.5) = -6 + 10 = 4. RHS: 3/(-0.5+2) + 5/(-0.5+3) = 3/1.5 + 5/2.5 = 2 + 2 = 4. Both sides are equal.
Example 2: Solve 2/(x+1) + 1/(x+2) = 2/(x+4) + 1/(x+5)
- Step 1: Pair identical numerators: 2(1/(x+1) - 1/(x+4)) + 1(1/(x+2) - 1/(x+5)) = 0.
- Step 2: Simplify: 2(3)/((x+1)(x+4)) + 1(3)/((x+2)(x+5)) = 0 → 6/(x² + 5x + 4) + 3/(x² + 7x + 10) = 0.
- Step 3: 2(x² + 7x + 10) + (x² + 5x + 4) = 0 → 3x² + 19x + 24 = 0.
- Step 4: x = (-19 ± √73)/6.
- Check: Verify the quadratic 3x² + 19x + 24 = 0 derived from 2(x² + 7x + 10) + (x² + 5x + 4) = 0. Its roots are x = (-19 ± √(361 - 288))/6 = (-19 ± √73)/6. Substituting these into the paired form confirms the zero sum.
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