Question 1209640: Find all solutions x to the equation
(x^4 + 2x^3 - 15x^2) - (4x^2 + 8x - 60) = x^4 + 10x^3 - 7x^2 - 5x - 3.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to find the solutions to the equation:
1. **Simplify both sides:**
Left side:
x⁴ + 2x³ - 15x² - 4x² - 8x + 60 = x⁴ + 2x³ - 19x² - 8x + 60
Right side:
x⁴ + 10x³ - 7x² - 5x - 3
2. **Set the equation to zero:**
Subtract the right side from the left side:
(x⁴ + 2x³ - 19x² - 8x + 60) - (x⁴ + 10x³ - 7x² - 5x - 3) = 0
x⁴ + 2x³ - 19x² - 8x + 60 - x⁴ - 10x³ + 7x² + 5x + 3 = 0
3. **Combine like terms:**
(x⁴ - x⁴) + (2x³ - 10x³) + (-19x² + 7x²) + (-8x + 5x) + (60 + 3) = 0
-8x³ - 12x² - 3x + 63 = 0
4. **Multiply by -1 (optional, but makes factoring easier):**
8x³ + 12x² + 3x - 63 = 0
5. **Try factoring:**
This cubic equation is a bit challenging to factor directly. Let's try the Rational Root Theorem. Possible rational roots are factors of 63 divided by factors of 8. Some possibilities are ±1, ±3, ±7, ±9, ±21, ±63, ±1/2, ±3/2, ±7/2, ±9/2, etc.
Trying x = 3/2:
8(3/2)³ + 12(3/2)² + 3(3/2) - 63 = 8(27/8) + 12(9/4) + 9/2 - 63 = 27 + 27 + 4.5 - 63 = 58.5 - 63 = -4.5. This is not zero.
Let's try synthetic division or a cubic solver. Using a calculator or software to solve cubic equations, we find one real root at x = 1.547 (approximately).
Since this is a cubic equation, it must have three roots (counting multiplicity).
Let's use a cubic solver. The approximate roots are:
x ≈ 1.54722687
The other two roots are complex conjugates.
Therefore, there is one real solution and two complex solutions to the given cubic equation. The approximate real solution is x ≈ 1.547.
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