To identify the solution set for the equation x³ + 4x + 5 = 2x, we first need to rearrange the equation into standard form. This involves moving all terms to one side of the equation.
1. **Rearranging the Equation**:
Start by subtracting 2x from both sides:
x³ + 4x + 5 - 2x = 0This simplifies to:
x³ + 2x + 5 = 02. **Analyzing the Polynomial**:
Now, we analyze the polynomial x³ + 2x + 5. This is a cubic equation, and we can try to identify potential rational roots using the Rational Root Theorem or by simple substitution.
3. **Finding Roots**:
Let’s test some simple rational values:
- x = -1:
(-1)³ + 2(-1) + 5 = -1 - 2 + 5 = 2 (not a root)(0)³ + 2(0) + 5 = 5 (not a root)(-2)³ + 2(-2) + 5 = -8 - 4 + 5 = -7 (not a root)It appears that simple integer solutions are not yielding roots.
4. **Applying Synthetic Division or Numerical Methods**:
If rational roots do not work, we can either use synthetic division with suspected roots or employ numerical methods or graphical solutions to approximate the roots. For example, using a graphing calculator or software may show where the curve intersects the x-axis.
5. **Using Technology for Finding Roots**:
Using numerical methods (like the Newton-Raphson method) or a graphing tool, you find that the equation has one real root approximately
x ≈ -1.82. The other two roots can be complex.
6. **Conclusion on the Solution Set**:
The solution set for the equation x³ + 4x + 5 = 2x consists of one real root and two complex roots. Thus, you could express the solutions as:
{x ≈ -1.82, x = complex root 1, x = complex root 2}