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 three methods:

• Graphical Method
• Algebraic Method

Graphical Method

In Graphical Method, we first draw the graph of the inequality and then find the solution of the given inequality using the graph. Lets 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 following three methods

• Factoring
• Quadratic Formula
• Completing the Square

Factoring

In Factoring, we split the given quadratic expression in product of its factors to find out the solution.

Example: Find out the solution for the inequality x² – 3x – 4 > 0 using Factoring Method

Solution:

Given inequality: x² – 3x – 4 > 0

Step 1: Factor the quadratic expression:

(x – 4)(x + 1) > 0

Step 2: Identify intervals based on the factors:

Interval I: x < -1

For x < -1, consider the factors (x – 4)(x + 1):

Pick a test point within the interval, say x = -2, substitute into (x – 4)(x + 1):

((-2) – 4)((-2) + 1) = (-6)(-1) = 6 > 0

The inequality holds for x < -1.

Interval II: -1 < x < 4

For -1 < x < 4, consider the factors (x – 4)(x + 1):

Pick a test point within the interval, say x = 0, substitute into (x – 4)(x + 1):

(0 – 4)(0 + 1) = (-4)(1) = -4 < 0

The inequality doesn’t hold for -1 < x < 4.

Interval III: x > 4

For x > 4, consider the factors (x – 4)(x + 1):

Pick a test point within the interval, say x = 5, substitute into (x – 4)(x + 1):

(5 – 4)(5 + 1) = (1)(6) = 6 > 0

The inequality holds for x > 4.

Therefore, the solution is x < -1 or x > 4.

Quadratic Formula

Use the quadratic formula to find the roots and determine the intervals where the expression is positive.

Example: Find the solution of the given inequality x² – 3x – 4 > 0 using Quadratic Formula

Solution:

Step 1: Apply the quadratic formula to find the roots:

x = [3 ± √((-3)² – 4×1×(-4))]/(2×1)

x = [3 ± √(25)]/2

x = [3 ± 5]/2

Roots: x = 4 or x = -1

Step 2: Test intervals to determine where the expression is positive:

Interval I: x < -1 → (x² – 3x – 4) > 0

For x < -1, consider the quadratic expression (x² – 3x – 4):

Choose a test point, say x = -2, substitute into (x² – 3x – 4):

((-2)² – 3(-2) – 4) = (4 + 6 – 4) = 6 > 0

The inequality holds for x < -1.

Interval II: -1 < x < 4 → (x² – 3x – 4) < 0

For -1 < x < 4, consider the quadratic expression (x² – 3x – 4):

Choose a test point, say x = 0, substitute into (x² – 3x – 4):

((0)² – 3(0) – 4) = (-4) < 0

The inequality doesn’t hold for -1 < x < 4.

Interval III: x > 4 → (x² – 3x – 4) > 0

For x > 4, consider the quadratic expression (x² – 3x – 4):

Choose a test point, say x = 5, substitute into (x² – 3x – 4):

((5)² – 3(5) – 4) = (25 – 15 – 4) = 6 > 0

The inequality holds for x > 4.

Therefore, the solution is x < -1 or x > 4.

Completing the Square

In completing the square method, convert the quadratic expression into a perfect square trinomial to solve the inequality.

Example: Solve the given inequality x² – 3x – 4 > 0 by using Completing the Square Method

Solution:

Step 1: Complete the square:

x² – 3x – 4 = (x – (3/2))² – 25/4

Step 2: Set up the inequality:

(x – (3/2))² – 25/4 > 0

Step 3: Find the intervals where the inequality holds:

This gives two cases:

Case 1: x – (3/2) > 5/2

x > 8/2

x > 4

Case 2: x – (3/2) < -5/2

x < -2/2

x < -1

Therefore, the solution is x < -1 or x > 4.

Quadratic inequalities are a type of algebraic inequality that involves quadratic expressions. A quadratic inequality looks like ax²+bx+c>0, ax²+bx+c<0, ax2+bx+c≥0 or ax2+bx+c≤, where a, b, and c are constants, and a≠0.

It is a fundamental concept in mathematics, particularly in algebra and calculus. They involve expressions containing quadratic polynomials and inequality signs. It is often required in solution sets that fulfill specific criteria. Lets know more about Quadratic Inequalities in detail below.