To solve the equation \( 2x^3 - 3x^2 + 8x - 9 = 3 \), we first simplify it by bringing all terms to one side of the equation:
\[ 2x^3 - 3x^2 + 8x - 9 - 3 = 0 \]
This simplifies to:
\[ 2x^3 - 3x^2 + 8x - 12 = 0 \]
Next, we can try to find rational roots using the Rational Root Theorem. The possible rational roots can be factors of the constant term (-12) divided by factors of the leading coefficient (2). The factors of -12 include ±1, ±2, ±3, ±4, ±6, ±12.
Testing some possible roots, let's check \( x = 2 \):
\[ 2(2)^3 - 3(2)^2 + 8(2) - 12 = 2(8) - 3(4) + 16 - 12 = 16 - 12 + 16 - 12 = 8 \quad \text{(not a root)} \]
Next, check \( x = 1.5 \):
\[ 2(1.5)^3 - 3(1.5)^2 + 8(1.5) - 12 \] Calculating: \[ = 2(3.375) - 3(2.25) + 12 - 12 = 6.75 - 6.75 + 12 - 12 = 0 \quad \text{(a root)} \]
So, \( x = 1.5 \) is a root.
Now we can factor the cubic polynomial using synthetic division or polynomial long division with \( x - 1.5 \).
Performing synthetic division:
1.5 | 2 -3 8 -12
| 3 3 16
------------------
2 0 11 4
The result of the synthetic division gives us \( 2x^2 + 11 \). Now the function is factored as:
\[ (x - 1.5)(2x^2 + 11) = 0 \]
The quadratic \( 2x^2 + 11 = 0 \) has no real solutions because:
\[ 2x^2 = -11 \implies x^2 = -\frac{11}{2} \]
The only real solution we found is \( x = 1.5 \).
Thus, the complete set of real solutions to the equation \( 2x^3 - 3x^2 + 8x - 9 = 3 \) is:
\[ \boxed{1.5} \]