The complete solution set of the equation `|x^2-5x+6|+|x^2+12x+27|=|17x + 21 | ` is a.`x in [-9,3]` b.`x in [-3,2) cup (2,3] ` c.`x in [-9 , -3] cup [ 2,3]` d.`x in (-2,3)`

Options

A. `x in [-9,3]`
B. `x in [-3,2) cup (2,3] `
C. `x in [-9 , -3] cup [ 2,3]`
D. `x in (-2,3)`

Explanation

To solve the equation \( |x^2 - 5x + 6| + |x^2 + 12x + 27| = |17x + 21| \), we will break it down step by step.

### Step 1: Identify the expressions inside the absolute values
We have three expressions to analyze:
1. \( f(x) = x^2 - 5x + 6 \)
2. \( g(x) = x^2 + 12x + 27 \)
3. \( h(x) = 17x + 21 \)

### Step 2: Find the roots of the quadratic equations
To determine the intervals where the expressions change sign, we need to find the roots of \( f(x) \) and \( g(x) \).

**For \( f(x) = x^2 - 5x + 6 \):**
\[ f(x) = 0 \implies x^2 - 5x + 6 = 0 \]
Factoring gives:
\[ (x - 2)(x - 3) = 0 \implies x = 2, 3 \]

**For \( g(x) = x^2 + 12x + 27 \):**
\[ g(x) = 0 \implies x^2 + 12x + 27 = 0 \]
Using the quadratic formula:
\[ x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 1 \cdot 27}}{2 \cdot 1} = \frac{-12 \pm \sqrt{144 - 108}}{2} = \frac{-12 \pm \sqrt{36}}{2} = \frac{-12 \pm 6}{2} \]
This gives:
\[ x = -3 \quad \text{and} \quad x = -9 \]

### Step 3: Identify critical points
The critical points from the roots are:
- From \( f(x) \): \( x = 2, 3 \)
- From \( g(x) \): \( x = -3, -9 \)
- From \( h(x) \): \( 17x + 21 = 0 \implies x = -\frac{21}{17} \approx -1.24 \)

### Step 4: Determine the intervals
The critical points divide the number line into intervals:
1. \( (-\infty, -9) \)
2. \( (-9, -3) \)
3. \( (-3, -\frac{21}{17}) \)
4. \( (-\frac{21}{17}, 2) \)
5. \( (2, 3) \)
6. \( (3, \infty) \)

### Step 5: Test each interval
We will test points in each interval to determine the sign of each expression.

1. **Interval \( (-\infty, -9) \):** Test \( x = -10 \)
- \( f(-10) = 16 + 50 + 6 = 72 \) (positive)
- \( g(-10) = 100 - 120 + 27 = 7 \) (positive)
- \( h(-10) = -170 + 21 = -149 \) (negative)
- Result: \( |72| + |7| = 79 \) vs \( |-149| = 149 \) (not satisfied)

2. **Interval \( (-9, -3) \):** Test \( x = -6 \)
- \( f(-6) = 36 + 30 + 6 = 72 \) (positive)
- \( g(-6) = 36 - 72 + 27 = -9 \) (negative)
- \( h(-6) = -102 + 21 = -81 \) (negative)
- Result: \( |72| + |-9| = 81 \) vs \( |-81| = 81 \) (satisfied)

3. **Interval \( (-3, -\frac{21}{17}) \):** Test \( x = -2 \)
- \( f(-2) = 4 + 10 + 6 = 20 \) (positive)
- \( g(-2) = 4 - 24 + 27 = 7 \) (positive)
- \( h(-2) = -34 + 21 = -13 \) (negative)
- Result: \( |20| + |7| = 27 \) vs \( |-13| = 13 \) (not satisfied)

4. **Interval \( (-\frac{21}{17}, 2) \):** Test \( x = 0 \)
- \( f(0) = 6 \) (positive)
- \( g(0) = 27 \) (positive)
- \( h(0) = 21 \) (positive)
- Result: \( |6| + |27| = 33 \) vs \( |21| = 21 \) (not satisfied)

5. **Interval \( (2, 3) \):** Test \( x = 2.5 \)
- \( f(2.5) = 6.25 - 12.5 + 6 = -0.25 \) (negative)
- \( g(2.5) = 6.25 + 30 + 27 = 63.25 \) (positive)
- \( h(2.5) = 42.5 + 21 = 63.5 \) (positive)
- Result: \( |-0.25| + |63.25| = 63.5 \) vs \( |63.5| = 63.5 \) (satisfied)

6. **Interval \( (3, \infty) \):** Test \( x = 4 \)
- \( f(4) = 16 - 20 + 6 = 2 \) (positive)
- \( g(4) = 16 + 48 + 27 = 91 \) (positive)
- \( h(4) = 68 + 21 = 89 \) (positive)
- Result: \( |2| + |91| = 93 \) vs \( |89| = 89 \) (not satisfied)

### Step 6: Combine the results
The intervals where the equation holds true are:
- From \( (-9, -3) \)
- From \( (2, 3) \)

Thus, the complete solution set is:
\[ x \in [-9, -3] \cup [2, 3] \]

### Final Answer
The correct option is:
**c. \( x \in [-9, -3] \cup [2, 3] \)**