Quadratic Inequalities

A quadratic inequality involves comparing a quadratic expression ax² + bx + c, where a ≠ 0 to a number or another polynomial (degree ≤ 2), using >, <, ≥, or ≤. Unlike a quadratic equation (which equals something), inequalities yield ranges of solutions. Examples are:

• x² + x − 1 > 0

• 2x² − 5x − 2

• x² + 2x − 1 < 0

Quadratic inequalities compare a quadratic expression to zero, using inequality symbols: greater than, less than, greater than or equal to, and less than or equal to. They are solved by finding roots and testing the signs of the expression over different intervals.

Step 1 -  Rewrite the equation to express the inequality:

Example: x² − 5x + 6 > 0.

Step 2 - Find roots by factoring or formula:

(x−2)(x−3)>0 → roots: 2, 3.

Step 3 - Create intervals:

(−∞,2),  (2,3),  (3,∞).

Step 4 - Test each interval by plugging in a sample value:

• x = 0: positive → true
   
• x = 2.5: negative → false
   
• x = 4: positive → true

Step 5. Write a solution:

All x∈ (−∞,2)  (3,∞).

The symbols > and < (greater than, less than) replace “=” in a quadratic equation to form a quadratic inequality. The format is
ax² + bx + c > 0   or   ax² + bx + c < 0.

• ( ) → Open brackets

• [ ] → Closed brackets

• o → Open value (x cannot take this value)

• • → Closed value (x can take this value)

Examples:

• (-1, 1) → x cannot take -1 and 1

• [-1, 1) → x can take -1 but not 1

• (-1, 1] → x cannot take -1 but can take 1

• [-1, 1] → x can take both -1 and 1

Different methods used to solve quadratic inequalities are listed below:

1. Rewrite as ax² + bx + c ≷ 0.
    
2. Factor & find roots.
    
3. Divide the number line into intervals.
    
4. Test each interval.

Example:

X² − 5x + 6 > 0  ⟹  (x − 2)(x − 3) > 0

(−∞,2), (2,3), (3,∞).

Testing the values in the expression, we get, x = 0: true; x = 2.5: false; x = 4: true.

Solution: x∈(−∞,2)∪(3,∞).

1. Factor the expression.
    
2. Mark the roots on a number line.
    
3. Determine the sign of each factor in each region (alternating sign from rightmost interval).
    
4. Combine to find where the product matches the inequality.

Example:

(x − 3)(x + 2) < 0.

x - 3 = 0, x = 3

x + 2 = 0, x = -2

So, the expression changes sign between x = -2 and x = 3

Now, set intervals using roots to divide the number line:

(-∞,-2), (-2,3), (3,∞)

Test each interval

x = -3: (x - 3) (x + 2) = (+)

x = 0: (x - 3) (x + 2) = (-)

x = 4: (x - 3) (x + 2) = (+)

The only region where the expression is negative is (-2, 3).
So, x∈(-2,3)

Solution: x∈(-2,3)

1. Rewrite as y = ax² + bx + c and y ≷ 0.
    
2. Sketch the parabola (upwards if a>0, down if a<0)
    
3. Read off where the parabola is above/below the x-axis.

Example:

x² + 5x + 6 ≥ 0.

The graph crosses at x = −2 and −3 and opens upward since the linear coefficient is positive.

So, the expression is ≥ 0 outside [−3,−2].

Solution: x∈(−∞, −3]∪[−2,∞).

The quadratic inequalities model helps us understand and solve real-world problems in the fields of physics, engineering, economics, and biology. Some of its applications are listed below:

• Bridge design: This is useful to ensure the structural integrity of bridges. For example, civil engineers need to examine the maximum load a bridge can bear to determine the range of weight distribution that is safe. This helps in preventing breakdown.

• Profit Margins in Business: Businesses use Quadratic Inequalities for modelling and analyzing the business profit margins. For example, a company uses it to decide the range of product prices that will create a desired profit level.

• Radio Telescope Design: Radio telescopes use parabolic mirrors to focus incoming radio waves. This helps design the shape and dimensions of these mirrors to ensure optimal signal reception and focusing. 

• Determining Safe Speed: This is useful in transportation, as it helps to find the Maximum safe speed on a curved and dangerous road map. Making sure that the vehicle can navigate the curve without skidding or losing control.

• Area Calculation: This can be used to find the dimensions of a shape, like a rectangle, that meets a specific area requirement. For example, to find the dimensions of a garden or a plot that has the minimum required area.