#### What is a Quadratic Inequality?
A **quadratic inequality** is an inequality that involves a quadratic expression and can be written in one of the following forms:
1. **Quadratic Less Than Zero:** `ax^2 + bx + c < 0`
2. **Quadratic Greater Than Zero:** `ax^2 + bx + c > 0`
3. **Quadratic Less Than or Equal To Zero:** `ax^2 + bx + c ≤ 0`
4. **Quadratic Greater Than or Equal To Zero:** `ax^2 + bx + c ≥ 0`
where `a`, `b`, and `c` are constants, and `a` is not zero.
#### Steps to Solve a Quadratic Inequality
Here’s a step-by-step guide:
1. **Rewrite the Inequality in Standard Form**
Ensure the quadratic expression is in standard form, which is `ax^2 + bx + c` with the inequality sign.
2. **Find the Roots of the Quadratic Equation**
Solve the corresponding quadratic equation `ax^2 + bx + c = 0` to find the roots. Use the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
- **`b^2 - 4ac`** is called the discriminant.
- **Positive Discriminant**: Two distinct real roots.
- **Zero Discriminant**: One real root (a repeated root).
- **Negative Discriminant**: No real roots.
3. **Plot the Roots on a Number Line**
The roots divide the number line into intervals.
4. **Test Each Interval**
Choose a test point from each interval and substitute it into the original inequality to see if the inequality holds.
5. **Write the Solution**
Write the solution in interval notation, representing the ranges of `x` that satisfy the inequality.
#### Example: Solving a Quadratic Inequality
Solve the inequality:
x^2 - 3x - 4 < 0
**Step 1: Rewrite the Inequality**
The inequality is in standard form.
**Step 2: Find the Roots**
Solve `x^2 - 3x - 4 = 0` using the quadratic formula:
x = [-(-3) ± sqrt((-3)^2 - 4*1*(-4))] / (2*1)
= [3 ± sqrt(9 + 16)] / 2
= [3 ± sqrt(25)] / 2
= [3 ± 5] / 2
So the roots are:
x = (3 + 5) / 2 = 4
x = (3 - 5) / 2 = -1
**Step 3: Plot the Roots**
The roots are `x = -1` and `x = 4`. These points divide the number line into three intervals: `(-∞, -1)`, `(-1, 4)`, and `(4, ∞)`.
**Step 4: Test Each Interval**
- For `x < -1` (e.g., `x = -2`):
Substitute `x = -2` into `x^2 - 3x - 4`:
(-2)^2 - 3*(-2) - 4 = 4 + 6 - 4 = 6 > 0
This interval does not satisfy `x^2 - 3x - 4 < 0`.
- For `-1 < x < 4` (e.g., `x = 0`):
Substitute `x = 0` into `x^2 - 3x - 4`:
0^2 - 3*0 - 4 = -4 < 0
This interval satisfies `x^2 - 3x - 4 < 0`.
- For `x > 4` (e.g., `x = 5`):
Substitute `x = 5` into `x^2 - 3x - 4`:
5^2 - 3*5 - 4 = 25 - 15 - 4 = 6 > 0
This interval does not satisfy `x^2 - 3x - 4 < 0`.
**Step 5: Write the Solution**
The solution to `x^2 - 3x - 4 < 0` is the interval where the inequality holds true:
(-1, 4)
#### Summary
To solve a quadratic inequality:
1. Rewrite it in standard form.
2. Find the roots using the quadratic formula.
3. Plot the roots on a number line to create intervals.
4. Test each interval to see where the inequality holds.
5. Write the solution in interval notation.
Understanding these steps helps in solving quadratic inequalities and applying these concepts in various mathematical problems.
