Which of the following is a complete set of real solutions to the equation 2x^3 - 3x^2 + 8x - 9 = 3?

A. No real solutions
B. -1.5, 2, -2
C. 1.5, 2, -2
D. 1.5

1 answer

To solve the equation \( 2x^3 - 3x^2 + 8x - 9 = 3 \), we start by rearranging it to set it to zero:

\[ 2x^3 - 3x^2 + 8x - 9 - 3 = 0 \]

This simplifies to:

\[ 2x^3 - 3x^2 + 8x - 12 = 0 \]

Next, we will try to find the roots of the polynomial \( 2x^3 - 3x^2 + 8x - 12 \). One method to find rational roots is to use the Rational Root Theorem, which suggests trying factors of the constant term (-12) divided by factors of the leading coefficient (2).

The potential rational roots we can try are:

\[ \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12, \pm \frac{1}{2}, \pm \frac{3}{2}, \pm \frac{4}{2}, \pm \frac{6}{2}, \pm \frac{12}{2} \]

We will start testing some of the simpler rational candidates:

  1. Evaluating \( x = 2 \):

\[ 2(2)^3 - 3(2)^2 + 8(2) - 12 = 2(8) - 3(4) + 16 - 12 = 16 - 12 + 16 - 12 = 8 \quad (\text{not a root}) \]

  1. Evaluating \( x = 1.5 \):

\[ 2(1.5)^3 - 3(1.5)^2 + 8(1.5) - 12 = 2(3.375) - 3(2.25) + 12 - 12 = 6.75 - 6.75 + 12 - 12 = 0 \quad (\text{is a root}) \]

  1. Evaluating \( x = -2 \):

\[ 2(-2)^3 - 3(-2)^2 + 8(-2) - 12 = 2(-8) - 3(4) - 16 - 12 = -16 - 12 - 16 - 12 = -56 \quad (\text{not a root}) \]

Now, we found that \( x = 1.5 \) is indeed a root. Next, we can use synthetic division or polynomial long division to divide \( 2x^3 - 3x^2 + 8x - 12 \) by \( (x - 1.5) \):

To facilitate calculations, we can express \( 1.5 \) as \( \frac{3}{2} \) and perform synthetic division using \( \frac{3}{2} \):

The synthetic division yields:

\[ 2 \quad -3 \quad 8 \quad -12 \]

  1. Bring down the 2.
  2. Multiply by \( \frac{3}{2} \), and get \( 3 \).
  3. Add: \( -3 + 3 = 0 \).
  4. Multiply by \( \frac{3}{2} \), and get \( 0 \).
  5. Add: \( 8 + 0 = 8 \).
  6. Multiply by \( \frac{3}{2} \), and get \( 12 \).
  7. Add: \( -12 + 12 = 0 \).

The quadratic polynomial is \( 2x^2 + 0x + 8 \), which simplifies to:

\[ 2x^2 + 8 = 0 \implies x^2 = -4 \implies x = \pm 2i \]

Thus, the complete set of real solutions to the equation \( 2x^3 - 3x^2 + 8x - 12 = 0 \) is:

\[ { 1.5 } \]

Since the only real solution is \( 1.5 \), the answer is:

D. 1.5.