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Answer :
Sure! Let's solve the quadratic inequality [tex]\( x^2 - 3x - 54 > 0 \)[/tex] step-by-step.
### Step 1: Find the Roots of the Quadratic Equation
First, we need to find the roots of the quadratic equation [tex]\( x^2 - 3x - 54 = 0 \)[/tex]. These roots will help us determine the critical points where the inequality might change.
The quadratic equation is [tex]\( x^2 - 3x - 54 = 0 \)[/tex].
To find the roots, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our quadratic equation [tex]\( x^2 - 3x - 54 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -3 \)[/tex]
- [tex]\( c = -54 \)[/tex]
Plug these values into the quadratic formula:
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-54)}}{2(1)} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 + 216}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{225}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm 15}{2} \][/tex]
So the roots are:
[tex]\[ x = \frac{3 + 15}{2} = 9 \][/tex]
[tex]\[ x = \frac{3 - 15}{2} = -6 \][/tex]
### Step 2: Determine the Intervals
The roots [tex]\(-6\)[/tex] and [tex]\(9\)[/tex] divide the number line into three intervals:
1. [tex]\( x < -6 \)[/tex]
2. [tex]\( -6 < x < 9 \)[/tex]
3. [tex]\( x > 9 \)[/tex]
### Step 3: Test Each Interval
We need to determine where the inequality [tex]\( x^2 - 3x - 54 > 0 \)[/tex] holds true.
1. Interval [tex]\( x < -6 \)[/tex]
- Choose [tex]\( x = -7 \)[/tex] (a test point in this interval)
- Calculate [tex]\( (-7)^2 - 3(-7) - 54 = 49 + 21 - 54 = 16 \)[/tex]
- Since [tex]\( 16 > 0 \)[/tex], this interval satisfies the inequality.
2. Interval [tex]\(-6 < x < 9 \)[/tex]
- Choose [tex]\( x = 0 \)[/tex] (a test point in this interval)
- Calculate [tex]\( (0)^2 - 3(0) - 54 = -54 \)[/tex]
- Since [tex]\( -54 < 0 \)[/tex], this interval does not satisfy the inequality.
3. Interval [tex]\( x > 9 \)[/tex]
- Choose [tex]\( x = 10 \)[/tex] (a test point in this interval)
- Calculate [tex]\( (10)^2 - 3(10) - 54 = 100 - 30 - 54 = 16 \)[/tex]
- Since [tex]\( 16 > 0 \)[/tex], this interval satisfies the inequality.
### Step 4: Write the Solution
The solution to the inequality [tex]\( x^2 - 3x - 54 > 0 \)[/tex] is where the inequality holds true:
So, [tex]\( x \)[/tex] is in the intervals [tex]\( x < -6 \)[/tex] or [tex]\( x > 9 \)[/tex].
Therefore, the solution to the inequality is:
[tex]\[ x < -6 \quad \text{or} \quad x > 9 \][/tex]
So, the correct answer is:
```plaintext
x < -6 or x > 9
```
### Step 1: Find the Roots of the Quadratic Equation
First, we need to find the roots of the quadratic equation [tex]\( x^2 - 3x - 54 = 0 \)[/tex]. These roots will help us determine the critical points where the inequality might change.
The quadratic equation is [tex]\( x^2 - 3x - 54 = 0 \)[/tex].
To find the roots, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For our quadratic equation [tex]\( x^2 - 3x - 54 \)[/tex], the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -3 \)[/tex]
- [tex]\( c = -54 \)[/tex]
Plug these values into the quadratic formula:
[tex]\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-54)}}{2(1)} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{9 + 216}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm \sqrt{225}}{2} \][/tex]
[tex]\[ x = \frac{3 \pm 15}{2} \][/tex]
So the roots are:
[tex]\[ x = \frac{3 + 15}{2} = 9 \][/tex]
[tex]\[ x = \frac{3 - 15}{2} = -6 \][/tex]
### Step 2: Determine the Intervals
The roots [tex]\(-6\)[/tex] and [tex]\(9\)[/tex] divide the number line into three intervals:
1. [tex]\( x < -6 \)[/tex]
2. [tex]\( -6 < x < 9 \)[/tex]
3. [tex]\( x > 9 \)[/tex]
### Step 3: Test Each Interval
We need to determine where the inequality [tex]\( x^2 - 3x - 54 > 0 \)[/tex] holds true.
1. Interval [tex]\( x < -6 \)[/tex]
- Choose [tex]\( x = -7 \)[/tex] (a test point in this interval)
- Calculate [tex]\( (-7)^2 - 3(-7) - 54 = 49 + 21 - 54 = 16 \)[/tex]
- Since [tex]\( 16 > 0 \)[/tex], this interval satisfies the inequality.
2. Interval [tex]\(-6 < x < 9 \)[/tex]
- Choose [tex]\( x = 0 \)[/tex] (a test point in this interval)
- Calculate [tex]\( (0)^2 - 3(0) - 54 = -54 \)[/tex]
- Since [tex]\( -54 < 0 \)[/tex], this interval does not satisfy the inequality.
3. Interval [tex]\( x > 9 \)[/tex]
- Choose [tex]\( x = 10 \)[/tex] (a test point in this interval)
- Calculate [tex]\( (10)^2 - 3(10) - 54 = 100 - 30 - 54 = 16 \)[/tex]
- Since [tex]\( 16 > 0 \)[/tex], this interval satisfies the inequality.
### Step 4: Write the Solution
The solution to the inequality [tex]\( x^2 - 3x - 54 > 0 \)[/tex] is where the inequality holds true:
So, [tex]\( x \)[/tex] is in the intervals [tex]\( x < -6 \)[/tex] or [tex]\( x > 9 \)[/tex].
Therefore, the solution to the inequality is:
[tex]\[ x < -6 \quad \text{or} \quad x > 9 \][/tex]
So, the correct answer is:
```plaintext
x < -6 or x > 9
```
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