Quadratic Inequality Solver
Solve Quadratic Inequalities
Enter a quadratic inequality (example: x² − 5x + 6 > 0) and click
Solve to understand the solution process.
Quadratic Inequalities – Complete Explanation
A quadratic inequality is an inequality that involves a quadratic expression, usually written in the form:
ax² + bx + c > 0 ax² + bx + c ≥ 0 ax² + bx + c < 0 ax² + bx + c ≤ 0
Instead of finding one solution, we determine the range of values for which the quadratic expression is positive, negative, or zero.
This comprehensive guide will take you through the complete process of solving quadratic inequalities, from basic concepts to advanced applications.
1. Key Idea Behind Quadratic Inequalities
Quadratic expressions form parabolas when graphed. Solving a quadratic inequality means finding where the parabola is above the x-axis (> 0), below the x-axis (< 0), or touching it (≥ 0 or ≤ 0).
The graph of a quadratic function y = ax² + bx + c is a parabola that:
- Opens upward if a > 0
- Opens downward if a < 0
- Crosses the x-axis at the roots (solutions of ax² + bx + c = 0)
2. General Steps to Solve Quadratic Inequalities
- Rewrite the inequality so one side equals 0
- Solve the related quadratic equation (find the roots)
- Identify the critical values (roots)
- Divide the number line into intervals based on these roots
- Test each interval by picking a point and evaluating the expression
- Determine which intervals satisfy the inequality
- Write the solution in interval notation or inequality form
3. Step-by-Step Examples
Example 1: Greater Than Zero (Positive)
x² − 5x + 6 > 0
Step 1: Factor the quadratic
(x − 2)(x − 3) > 0
Step 2: Find critical points (roots)
x = 2, x = 3
Step 3: Test intervals
Interval 1: x < 2 (test x = 0) (0-2)(0-3) = (-)(-) = (+) → positive Interval 2: 2 < x < 3 (test x = 2.5) (2.5-2)(2.5-3) = (0.5)(-0.5) = (-) → negative Interval 3: x > 3 (test x = 4) (4-2)(4-3) = (2)(1) = (+) → positive
Step 4: Determine solution for > 0
The expression is positive when x < 2 or x > 3
Final solution:
x < 2 or x > 3 Interval notation: (-∞, 2) ∪ (3, ∞)
Example 2: Less Than or Equal To Zero
x² − 4x + 3 ≤ 0
Step 1: Factor
(x − 1)(x − 3) ≤ 0
Step 2: Critical points
x = 1, x = 3
Step 3: Test intervals
x < 1 (test x = 0): ( - )( - ) = (+) → positive 1 < x < 3 (test x = 2): ( + )( - ) = (-) → negative x > 3 (test x = 4): ( + )( + ) = (+) → positive
Step 4: Solution for ≤ 0
The expression is negative or zero between roots, including endpoints
Final solution:
1 ≤ x ≤ 3 Interval notation: [1, 3]
Example 3: Leading Coefficient Negative
−x² + 2x + 3 > 0
Step 1: Multiply both sides by −1 and reverse the inequality
x² − 2x − 3 < 0
Step 2: Factor
(x − 3)(x + 1) < 0
Step 3: Critical points
x = 3, x = -1
Step 4: Test intervals
x < -1 (test x = -2): (-)(-) = (+) → positive -1 < x < 3 (test x = 0): (-)(+) = (-) → negative x > 3 (test x = 4): (+)(+) = (+) → positive
Step 5: Solution for < 0
-1 < x < 3
Example 4: Perfect Square
x² − 6x + 9 > 0
Step 1: Factor
(x − 3)² > 0
Step 2: Critical point
x = 3 (double root)
Step 3: Test intervals
x < 3: (negative)² = (+) → positive x > 3: (positive)² = (+) → positive x = 3: (0)² = 0 → not > 0
Final solution:
x < 3 or x > 3 (all real numbers except 3) Interval notation: (-∞, 3) ∪ (3, ∞)
Example 5: No Real Roots
x² + 2x + 5 > 0
Step 1: Check discriminant
Δ = 4 - 20 = -16 < 0 → no real roots
Step 2: Test one point (x = 0)
0² + 0 + 5 = 5 > 0
Step 3: Since parabola opens upward (a > 0) and is always positive:
All real numbers satisfy the inequality
Final solution: All real numbers (ℝ)
4. Using a Sign Chart
A sign chart is a systematic way to organize interval testing:
Interval | x < 2 | 2 < x < 3 | x > 3 ----------------------------------------------- (x-2) | - | + | + (x-3) | - | - | + ----------------------------------------------- Product | + | - | +
Steps to create a sign chart:
- Mark critical points on a number line
- Divide into intervals
- Test each factor's sign in each interval
- Multiply signs to determine overall sign
- Shade intervals that satisfy the inequality
5. Graphical Interpretation
- Parabola opens upward (a > 0): Positive outside roots, negative between roots
- Parabola opens downward (a < 0): Positive between roots, negative outside roots
- Include roots if inequality uses ≤ or ≥
- Exclude roots if inequality uses < or >
Graphing the parabola helps visualize the solution:
- Above x-axis → > 0
- Below x-axis → < 0
- On or above → ≥ 0
- On or below → ≤ 0
6. Special Cases
Case 1: Double Root (Perfect Square)
If the quadratic factors as (x - r)², the expression is always ≥ 0. The inequality (x - r)² > 0 has solution all real numbers except r.
Case 2: No Real Roots (Discriminant < 0)
If the discriminant is negative and a > 0, the expression is always positive. If a < 0, it's always negative.
Case 3: Always Zero
If all coefficients are zero, the expression is always 0.
7. Common Mistakes to Avoid
- Forgetting to reverse the inequality when multiplying or dividing by a negative number
- Using roots as the solution instead of intervals
- Not testing intervals correctly (using wrong test points)
- Ignoring ≤ or ≥ symbols when including/excluding endpoints
- Mixing up "and" vs "or" in compound solutions
- Forgetting to check the leading coefficient sign for parabola direction
- Incorrect factoring leading to wrong critical points
8. Practice Problems with Solutions
Problem 1: Solve x² − 9 ≥ 0
Solution: (x-3)(x+3) ≥ 0 → x ≤ -3 or x ≥ 3
Check: x = -4 works (16-9=7≥0); x = 0 doesn't work (-9≥0? No)
Problem 2: Solve x² + 2x − 8 < 0
Solution: (x+4)(x-2) < 0 → -4 < x < 2
Check: x = 0 works (-8<0); x = 3 doesn't work (9+6-8=7<0? No)
Problem 3: Solve −2x² + 8x − 6 ≥ 0
Solution: Divide by -2 and reverse: x² - 4x + 3 ≤ 0 → (x-1)(x-3) ≤ 0 → 1 ≤ x ≤ 3
Check: x = 2 works; x = 0 doesn't work
Problem 4: Solve x² + 4x + 4 > 0
Solution: (x+2)² > 0 → x ≠ -2 (all real numbers except -2)
Check: x = -1 works (1>0); x = -2 doesn't work (0>0? No)
Problem 5: Solve 2x² + 8 > 0
Solution: Discriminant = 0 - 64 = -64 < 0, a=2>0 → always positive → all real numbers
Check: Any x works (2x²+8 is always ≥8 > 0)
9. Real-World Applications
Quadratic inequalities appear in many real-world scenarios:
- Projectile motion: When is a projectile above a certain height? (h(t) > 20)
- Profit and revenue optimization: When is profit positive? (P(x) > 0)
- Engineering design constraints: Stress must be less than material strength (σ < σ_max)
- Safety and tolerance limits: Temperature must be within safe range (T_min ≤ T ≤ T_max)
- Economics and modeling: Supply and demand curves
- Physics: When is kinetic energy greater than potential energy?
- Sports: When does a ball stay in bounds?
Example: Projectile Motion
A ball is thrown upward with height h(t) = -16t² + 64t + 5 feet. When is the ball above 20 feet?
-16t² + 64t + 5 > 20 -16t² + 64t - 15 > 0 16t² - 64t + 15 < 0
Solve to find the time interval when the ball is above 20 feet.
10. Tips for Mastering Quadratic Inequalities
- Always factor first when possible
- Draw a number line and mark critical points
- Test one point in each interval
- Remember that > and < exclude endpoints, ≥ and ≤ include them
- For "and" compound inequalities, take intersection
- For "or" compound inequalities, take union
- Check your answer by substituting a value from each interval
- Practice with all inequality types (>, ≥, <, ≤)
- Graph the parabola to visualize the solution
- Don't forget to reverse signs when multiplying by negatives
11. Connection to Higher Mathematics
Quadratic inequalities are the foundation for:
- Polynomial inequalities (degree 3 and higher)
- Rational inequalities (fractions with variables)
- Optimization problems (finding maximum/minimum under constraints)
- Calculus (determining where functions are increasing/decreasing)
- Linear programming (feasible regions with quadratic constraints)
12. Final Thoughts
Quadratic inequalities combine algebra, graphing, and logical reasoning. Mastering them helps you understand how quadratic functions behave over different intervals and prepares you for more advanced mathematical concepts.
The key to success is practice with many different types—some with distinct roots, some with double roots, some with no real roots. Each case teaches you something new about the relationship between algebraic expressions and their graphs.
Use the solver above to guide your steps and check your work, but always practice manually to gain confidence and avoid common mistakes. With consistent practice, solving quadratic inequalities will become intuitive and automatic.