A quadratic inequality is an inequality that involves a quadratic expression of the form ax² + bx + c (where a ≠ 0) and compares it with zero, a constant, or another expression using symbols like >, <, ≥, or ≤. It represents the set of values of x for which the inequality is true.
Example
x² − 4x + 3 ≥ 0
2x² + x − 5 < 0
x² − 9 > 0
Types of Quadratic Inequalities
Quadratic inequalities can be expressed in the following standard forms:
ax2 + bx + c > 0 (The quadratic expression is positive.)
ax2 + bx + c < 0 (The quadratic expression is negative)
ax2 + bx + c ≥ 0 (The quadratic expression is non-negative)
ax2 + bx + c ≤ 0 (The quadratic expression is non-positive.)
Steps to solve a Quadratic Inequality
A quadratic inequality can be solved using either factorization or the quadratic formula.
Steps:
Step 1. Write in standard form Bring the inequality to the form: ax² + bx + c > 0, < 0, ≥ 0, or ≤ 0
Step 2. Find the roots Solve ax² + bx + c = 0
Factorise if possible, or
Use the quadratic formula
Step 3. Mark critical points Plot the roots on a number line. These points divide the number line into intervals.
Step 4. Test intervals Pick a value from each interval and substitute into the inequality to check if it satisfies the condition.
Step 5. Write the solution Select the intervals where the inequality is true and express the answer in inequality or interval notation.
Methods to Solve Quadratic Inequalities
Solving quadratic inequalities means finding the value of x for which it satisfies the given inequality. We can solve a quadratic inequality using two main methods:
Graphical Method
In the graphical method, we first draw the graph of the inequality and then find the solution of the given inequality using the graph. Let's see how to solve quadratic inequalities using the graphical method with the help of an example.
Example: x2 - 3x - 4 > 0
Solution:
We can solve the above example using graphical methods
Step 1: Plotting the graph of the quadratic function y = x2 - 3x - 4
Finding the x-intercepts: Given quadratic function: y = x2 - 3x - 4
To find the roots, solve for y = 0: x2 - 3x - 4 = 0
Factorizing the equation or using the quadratic formula: x2 - 4x + x - 4 = 0 x(x - 4) + 1(x - 4) = 0 (x - 4)(x + 1) = 0 Therefore, the roots are x = 4 and x = -1
Plotting the graph:
Step 2: Identifying regions where the graph lies above the x-axis (where y > 0)
We are looking for the regions where the quadratic function is positivee(y > 0). The function is y = x2 - 3x - 4. We know the roots are x = 4 and x = -1.
Step 3: Determining the x-values within these regions to obtain the solution set
Based on the graph, the regions where y > 0 are: - x < -1 - x > 4
Therefore, the solution set for x2 - 3x - 4 > 0 is x < -1 or x > 4.
This solution is derived from observing the parts of the graph where the quadratic function y = x2 - 3x - 4 is above the x-axis.
Algebraic Method
Algebraically, we can solve a quadratic inequality using the following three methods:
1. Factoring
In factoring, we split the given quadratic expression into the product of its factors to find out the solution.
Example: Find out the solution for the inequality x2 - 3x - 4 > 0 using the factoring method.
Step 2: Find intervals where the expression is non-positive: Interval I: -2/3 ≤ x ≤ 2 For -2/3 ≤ x ≤ 2, consider the factors (3x + 2)(x - 2):
Pick a test point within the interval, say x = 0, substitute into (3x + 2)(x - 2): (3 × 0 + 2)(0 - 2) = (2)(-2) = -4 ≤ 0 The inequality holds for -2/3 ≤ x ≤ 2.
Step 2: Find intervals where the expression is positive: Interval I: x > -3
For x > -3, consider the factor (x + 3)^2: For any value of x greater than -3, (x + 3)^2 will always be greater than zero because it represents a square value.
