To find the complete set of real solutions to the equation \(2x^3 - 3x^2 + 8x - 9 = 3\), we first rewrite the equation in standard form:
\[ 2x^3 - 3x^2 + 8x - 9 - 3 = 0 \] \[ 2x^3 - 3x^2 + 8x - 12 = 0 \]
Next, we can look for rational roots using the Rational Root Theorem or attempt synthetic division or substitution to find roots.
By substituting some of the given values, we can check for solutions:
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Testing \(x = 1.5\): \[ 2(1.5)^3 - 3(1.5)^2 + 8(1.5) - 12 = 2(3.375) - 3(2.25) + 12 - 12 \] \[ = 6.75 - 6.75 + 12 - 12 = 0 \] So \(x = 1.5\) is a solution.
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Testing \(x = 2\): \[ 2(2)^3 - 3(2)^2 + 8(2) - 12 = 2(8) - 3(4) + 16 - 12 \] \[ = 16 - 12 + 16 - 12 = 8 \neq 0 \] So \(x = 2\) is not a solution.
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Testing \(x = -2\): \[ 2(-2)^3 - 3(-2)^2 + 8(-2) - 12 = 2(-8) - 3(4) - 16 - 12 \] \[ = -16 - 12 - 16 - 12 = -56 \neq 0 \] So \(x = -2\) is not a solution.
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Testing \(x = -1.5\): \[ 2(-1.5)^3 - 3(-1.5)^2 + 8(-1.5) - 12 = 2(-3.375) - 3(2.25) - 12 + 12 \] \[ = -6.75 - 6.75 - 12 + 12 \neq 0 \] So \(x = -1.5\) is not a solution.
Next, since we found that \(1.5\) is a solution but \(2\) and \(-2\) are not, we should factor or simplify further to find other roots.
Using numerical or graphical methods can also help to find other possible solutions.
Through polynomial long division or synthetic division using \(x - 1.5\) should yield the remaining factors. If we find other rational or real roots (none from \(2\), \( -2\), or \(-1.5\)), then we confirm via a graph or by evaluating the sign changes or derivative tests.
For now, we suspect solutions are limited to \(x = 1.5\) based on our tests.
Thus, from the options given, the complete set of real solutions to the equation is:
1.5.
So the answer is 1.5.