Question

Which of the following is a complete set of real solutions to the equation 2x3−3x2+8x−9=3?(1 point)
Responses

1.5
1.5

no solutions
no solutions

1.5, 2, −2
1.5, 2, negative 2

−1.5, 2, −2

Answer 1

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:

\[
\boxed{1.5}
\]