`x ^(2) + 12 x-30`
What is the absolute value of the difference between the two solutions for x in the equation above ?

Options

A. `2 sqrt66`
B. 4
C. `6 sqrt2`
D. 12

Answer

To solve the quadratic inequality \( x^2 + 12x - 30 < 0 \) and find the absolute value of the difference between the two solutions for \( x \), we will follow these steps:

### Step 1: Identify coefficients
The quadratic equation is in the form \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 12 \)
- \( c = -30 \)

### Step 2: Apply the quadratic formula
The solutions for \( x \) can be found using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

### Step 3: Calculate the discriminant
First, we need to calculate the discriminant \( b^2 - 4ac \):
\[ b^2 = 12^2 = 144 \]
\[ 4ac = 4 \cdot 1 \cdot (-30) = -120 \]
Thus,
\[ b^2 - 4ac = 144 + 120 = 264 \]

### Step 4: Substitute into the quadratic formula
Now we substitute the values into the quadratic formula:
\[ x = \frac{-12 \pm \sqrt{264}}{2 \cdot 1} \]
\[ x = \frac{-12 \pm \sqrt{264}}{2} \]

### Step 5: Simplify the square root
Next, we simplify \( \sqrt{264} \):
\[ \sqrt{264} = \sqrt{4 \cdot 66} = 2\sqrt{66} \]
Thus, the solutions become:
\[ x = \frac{-12 \pm 2\sqrt{66}}{2} \]
\[ x = -6 \pm \sqrt{66} \]

### Step 6: Identify the two solutions
The two solutions for \( x \) are:
\[ x_1 = -6 - \sqrt{66} \]
\[ x_2 = -6 + \sqrt{66} \]

### Step 7: Calculate the absolute value of the difference
To find the absolute value of the difference between the two solutions:
\[ |x_2 - x_1| = |(-6 + \sqrt{66}) - (-6 - \sqrt{66})| \]
\[ = |(-6 + \sqrt{66} + 6 + \sqrt{66})| \]
\[ = |2\sqrt{66}| \]
\[ = 2\sqrt{66} \]

### Final Answer
The absolute value of the difference between the two solutions for \( x \) is \( 2\sqrt{66} \).

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