Complete set of solution of inequation `sqrt(3x^2+5x+7)-sqrt(3x^2+5x+2)gt1` is `(-a,-b)uu(-c,-d)` then find the value of a+b+c+d

Options

A. `4`
B. `3`
C. `2`
D. `1`

Answer

To solve the inequality \( \sqrt{3x^2 + 5x + 7} - \sqrt{3x^2 + 5x + 2} > 1 \), we will follow these steps:

### Step 1: Isolate one of the square roots
We can start by rewriting the inequality:
\[ \sqrt{3x^2 + 5x + 7} > 1 + \sqrt{3x^2 + 5x + 2} \]

### Step 2: Square both sides
To eliminate the square roots, we square both sides:
\[ 3x^2 + 5x + 7 > (1 + \sqrt{3x^2 + 5x + 2})^2 \]
Expanding the right-hand side:
\[ 3x^2 + 5x + 7 > 1 + 2\sqrt{3x^2 + 5x + 2} + (3x^2 + 5x + 2) \]
This simplifies to:
\[ 3x^2 + 5x + 7 > 3x^2 + 5x + 3 + 2\sqrt{3x^2 + 5x + 2} \]

### Step 3: Simplify the inequality
Subtract \(3x^2 + 5x + 3\) from both sides:
\[ 4 > 2\sqrt{3x^2 + 5x + 2} \]
Dividing both sides by 2 gives:
\[ 2 > \sqrt{3x^2 + 5x + 2} \]

### Step 4: Square both sides again
Square both sides again:
\[ 4 > 3x^2 + 5x + 2 \]
This can be rearranged to:
\[ 3x^2 + 5x - 2 < 0 \]

### Step 5: Factor the quadratic
We factor the quadratic:
\[ 3x^2 + 6x - x - 2 < 0 \]
Grouping gives:
\[ (3x - 1)(x + 2) < 0 \]

### Step 6: Find the critical points
Setting each factor to zero gives us the critical points:
\[ 3x - 1 = 0 \implies x = \frac{1}{3} \]
\[ x + 2 = 0 \implies x = -2 \]

### Step 7: Test intervals
We test the intervals created by the critical points: \((- \infty, -2)\), \((-2, \frac{1}{3})\), and \((\frac{1}{3}, \infty)\).

- For \(x < -2\) (e.g., \(x = -3\)):
\((3(-3) - 1)(-3 + 2) = (-10)(-1) > 0\)

- For \(-2 < x < \frac{1}{3}\) (e.g., \(x = 0\)):
\((3(0) - 1)(0 + 2) = (-1)(2) < 0\)

- For \(x > \frac{1}{3}\) (e.g., \(x = 1\)):
\((3(1) - 1)(1 + 2) = (2)(3) > 0\)

### Step 8: Conclusion
The solution to the inequality \(3x^2 + 5x - 2 < 0\) is:
\[ x \in (-2, \frac{1}{3}) \]

### Step 9: Express in required form
The complete set of solutions can be expressed as:
\[ (-2, -1) \cup (-\frac{2}{3}, \frac{1}{3}) \]
Thus, we have \(a = 2\), \(b = 1\), \(c = \frac{2}{3}\), and \(d = \frac{1}{3}\).

### Step 10: Calculate \(a + b + c + d\)
Now, we find:
\[ a + b + c + d = 2 + 1 + \frac{2}{3} + \frac{1}{3} = 3 + 1 = 4 \]

### Final Answer
The value of \(a + b + c + d\) is \(4\).