01.5.1 Introduction to Quadratic Inequalities
A quadratic inequality is an inequality that involves a quadratic expression.
Its standard forms are:
where:
-
\(a,b,c\) are real numbers.
-
\(a\neq0\).
-
\(x\) is the variable.
The condition \(a\neq0\) is essential because the expression must contain a nonzero \(x^2\) term.
Examples of quadratic inequalities
Example A
This is a quadratic inequality because the highest power of \(x\) is 2.
Example B
This is also a quadratic inequality.
Example C
This is a quadratic inequality even though the right-hand side is not zero.
To write it in standard form, subtract 8 from both sides:
We now have a quadratic expression compared with zero.
Quadratic equations vs. quadratic inequalities
Consider the quadratic equation:
Factorizing gives:
Therefore:
The equation has two solutions.
Now consider the quadratic inequality:
Its solution is:
The inequality has infinitely many real-number solutions between 2 and 3.
Key concept: A quadratic equation identifies where an expression equals zero. A quadratic inequality identifies where an expression is positive, negative, or zero, depending on the comparison symbol.
01.5.2 Understanding the Roots of a Quadratic Expression
Before solving a quadratic inequality, we often need to find the roots of the related quadratic equation.
A root, or zero, is a value of \(x\) that makes a quadratic expression equal to zero.
Consider:
To find its roots, set the expression equal to zero:
Factorize:
By the zero-product property:
or:
Therefore:
The roots are 3 and 4.
Why are roots important?
Roots divide the real number line into intervals where a quadratic expression has a constant sign.
For the expression:
The roots divide the number line into:
Within each of these open intervals, the expression is either always positive or always negative.
It cannot change sign inside an interval without passing through zero.
Important distinction
Finding the roots is not the final step when solving an inequality.
If a student finds:
and stops, the quadratic inequality has not yet been solved.
The student must determine which intervals satisfy the original inequality.
01.5.3 The Sign of a Quadratic Expression
Consider the general quadratic expression:
The sign of \(f(x)\) depends on:
-
The leading coefficient \(a\).
-
The roots of the quadratic equation.
-
The position of \(x\) relative to the roots.
Case 1: Positive leading coefficient
Suppose:
The parabola opens upward.
If the expression has two distinct real roots \(r_1<r_2\), then:
when:
And:
when:
In other words, the quadratic expression is positive outside the roots and negative between them.
Case 2: Negative leading coefficient
Suppose:
The parabola opens downward.
If there are two distinct real roots \(r_1<r_2\), the signs are reversed.
The expression is positive between the roots:
And negative outside the roots:
Summary of Signs
The following table applies when a quadratic expression
has two distinct real roots
r1 < r2.
| Leading coefficient | x < r1 | r1 < x < r2 | x > r2 |
|---|---|---|---|
| a > 0 | Positive | Negative | Positive |
| a < 0 | Negative | Positive | Negative |
Note: At both roots,
the quadratic expression equals zero.
Visual Interpretation
r₁
r₂
+
+
−
r₁
r₂
+
−
−
Important:
These sign patterns apply only when the quadratic
has two distinct real roots. Quadratics with one
repeated root or no real roots behave differently,
as explained later in this lesson.
01.5.4 Solving Quadratic Inequalities by Factorization
Factorization is one of the most effective methods for solving quadratic inequalities when the expression can be factored easily.
General solving procedure
Step 1: Write the inequality in standard form, with zero on one side.
Step 2: Factorize the quadratic expression.
Step 3: Find the roots by setting each factor equal to zero.
Step 4: Arrange the roots in ascending order.
Step 5: Determine the sign of the expression in each interval.
Step 6: Select the intervals satisfying the original inequality.
Step 7: Include or exclude the roots according to the inequality symbol.
Step 8: Express the final solution in interval notation.
Example 1: Solve a quadratic inequality with \(>\)
Solve:
Step 1. Factorize.
Find two numbers whose product is 6 and whose sum is −5.
Those numbers are −2 and −3.
Therefore:
Step 2. Find the roots.
Set each factor equal to zero:
Step 3. Divide the number line into intervals.
The critical values are 2 and 3.
The intervals are:
Step 4. Test each interval.
Choose one convenient value from each interval.
For \(x<2\), choose \(x=0\):
The expression is positive.
For \(2<x<3\), choose \(x=2.5\):
The expression is negative.
For \(x>3\), choose \(x=4\):
The expression is positive.
Step 5. Choose the correct intervals.
The original inequality requires:
Therefore, select the intervals where the product is positive.
Final answer:
x < 2 orx > 3
The empty box above is illustrative only; use the unboxed answer below in Tutor LMS until your MathJax AMS configuration is fixed.
Interval notation:
The roots are excluded because the inequality uses the strict symbol \(>\).
Example 2: Solve a quadratic inequality with \(\leq\)
Solve:
Factorize:
The roots are:
The quadratic opens upward because the leading coefficient is positive.
Therefore, the quadratic expression is negative between the roots and equals zero at the roots.
The inequality requires values that are negative or equal to zero.
Final answer:
Interval notation:
Notice that the roots are included because the inequality uses \(\leq\).
Comparing Examples 1 and 2
The same quadratic expression can produce different solution sets depending on the inequality symbol.
| Inequality | Solution |
|---|---|
| x2 − 5x + 6 > 0 | (−∞, 2) ∪ (3, ∞) |
| x2 − 5x + 6 ≥ 0 | (−∞, 2] ∪ [3, ∞) |
| x2 − 5x + 6 < 0 | (2, 3) |
| x2 − 5x + 6 ≤ 0 | [2, 3] |
These four relationships are especially important to understand.
01.5.5 Solving Quadratic Inequalities with a Negative Leading Coefficient
When the coefficient of \(x^2\) is negative, the parabola opens downward.
There are two useful approaches:
-
Analyze the downward-opening parabola directly.
-
Multiply the inequality by −1 to make the leading coefficient positive.
The second method is often easier for beginners.
Remember: Multiplying both sides of an inequality by −1 reverses the inequality sign.
Example 3: Negative leading coefficient
Solve:
Step 1. Multiply both sides by −1.
Reverse \(\geq\) to \(\leq\).
Step 2. Factorize.
Step 3. Find the roots.
In ascending order:
Step 4. Determine the correct interval.
The transformed quadratic has a positive leading coefficient, so it is negative between its two roots.
Because the inequality includes equality, the roots must also be included.
Final answer:
Interval notation:
Verification
Choose \(x=0\), which lies inside the solution interval.
Substitute into the original inequality:
This is true.
Now choose \(x=5\), which lies outside the solution interval:
This is false.
Therefore, the solution is consistent.
01.5.6 Quadratic Inequalities with Non-Unit Leading Coefficients
A quadratic inequality may have a leading coefficient other than 1 or −1.
For example:
We can still use factorization.
Example 4: Solving with \(a=2\)
Solve:
Step 1. Factorize.
Step 2. Find the roots.
From:
We obtain:
From:
We obtain:
Step 3. Determine the sign.
The leading coefficient is positive.
Therefore, the expression is negative between the two distinct roots.
Step 4. Write the final solution.
Interval notation:
The endpoints are excluded because the inequality is strict.
01.5.7 Using the Quadratic Formula
Not every quadratic expression can be factored easily using integers.
In such cases, we can find the roots using the quadratic formula.
For the quadratic equation:
where \(a\neq0\), the quadratic formula is:
After finding the real roots, we apply the same sign analysis used in factorization.
Example 5: Quadratic inequality with irrational roots
Solve:
Step 1. Identify the coefficients.
Step 2. Apply the quadratic formula.
Simplify:
Since:
We obtain:
Therefore, the roots are:
Step 3. Determine the solution interval.
The coefficient of \(x^2\) is positive, so the parabola opens upward.
The expression is negative between the roots.
Because the inequality includes equality, both roots are included.
Final answer:
Interval notation:
Exam tip: Keep answers in exact radical form unless the question specifically requests a decimal approximation.
For example, \(1+\sqrt2\) is more exact than its approximate value of 2.414.
01.5.8 The Discriminant and Special Cases
So far, our examples have involved quadratic expressions with two distinct real roots.
However, a quadratic equation can have:
-
Two distinct real roots.
-
One repeated real root.
-
No real roots.
The discriminant allows us to distinguish these cases.
For:
The discriminant is:
Case 1: Positive discriminant
When:
The quadratic equation has two distinct real roots.
The corresponding parabola crosses the x-axis twice.
The number line is divided into three intervals, and we can use the sign patterns established earlier.
For example:
The discriminant is:
Since \(\Delta>0\), there are two distinct real roots.
They are:
Case 2: Zero discriminant
When:
The quadratic equation has exactly one distinct real root, called a repeated root or double root.
The parabola touches the x-axis at this root without crossing it.
For example:
Factorize:
The only distinct root is:
The expression can be written as:
Since a square is always nonnegative:
for every real number \(x\).
Example 6: Repeated root with a strict inequality
Solve:
A square is positive for all real values except where the square equals zero.
The expression equals zero only when:
Therefore:
Interval notation:
Example 7: Repeated root with a non-strict inequality
Solve:
A square cannot be negative.
Therefore, the expression must equal zero.
Hence:
Final solution set:
This is a set containing exactly one value.
Case 3: Negative discriminant
When:
The quadratic equation has no real roots.
The corresponding parabola does not touch or cross the x-axis.
If \(a>0\), the quadratic expression is positive for every real value of \(x\).
If \(a<0\), the quadratic expression is negative for every real value of \(x\).
Example 8: Quadratic inequality with no real roots
Solve:
Method 1: Discriminant
Here:
Calculate:
Since \(\Delta<0\), the expression has no real roots.
The leading coefficient is positive, so the expression is positive for all real \(x\).
Therefore, it can never be less than zero.
Final answer:
∅
There is no real solution.
Method 2: Completing the square
Rewrite the original expression:
as:
Because:
We know that:
Therefore:
is impossible for real \(x\).
This confirms that the solution set is empty.
Example 9: All real numbers satisfy the inequality
Solve:
Using the same completed-square expression:
The left-hand side is always at least 1.
Therefore, the inequality is true for every real number.
Final answer:
Interval notation:
Discriminant Summary
| Discriminant | Real roots | Sign behavior |
|---|---|---|
| Δ > 0 | Two distinct roots | Sign changes at each root |
| Δ = 0 | One repeated root | Sign does not change at the root |
| Δ < 0 | No real roots | Sign is constant everywhere |
For Δ = 0,
the expression is zero at its repeated root and has the sign of
a everywhere else.
For Δ < 0,
its sign is always the sign of a.
01.5.9 Using Sign Charts and Number Lines
A sign chart is a method for identifying where a polynomial expression is positive or negative.
This is particularly useful when solving quadratic inequalities because it provides a systematic alternative to memorizing whether solutions lie inside or outside the roots.
Example 10: Solving with a sign chart
Solve:
Step 1. Find the roots.
⟹
x = −1
⟹
x = 4
Step 2. Identify the intervals.
Step 3. Test the sign in each interval.
For \(x<-1\), choose \(x=-2\).
The product is positive.
For \(-1<x<4\), choose \(x=0\).
The product is negative.
For \(x>4\), choose \(x=5\).
The product is positive.
Completed Sign Chart
| Interval | Test value | Sign |
|---|---|---|
| (−∞, −1) | −2 | Positive (+) |
| (−1, 4) | 0 | Negative (−) |
| (4, ∞) | 5 | Positive (+) |
The inequality asks for values where the product is less than zero.
Therefore, choose the interval with a negative sign.
Final answer:
Interval notation:
Understanding the number line
Understanding the Number Line
(x + 1)(x − 4) < 0
−2
−1
0
1
2
3
4
5
Illustrative number line: the highlighted green region
is the open interval
(−1, 4).
The two endpoints are excluded.
Important observation
For a quadratic with two distinct real roots, the sign cannot change without passing through a root.
Therefore, testing one value in each interval is sufficient to determine the sign throughout that interval.
For quadratics with a repeated root, the sign does not change when crossing that root.
01.5.10 Solving Quadratic Inequalities with Additional Conditions
Sometimes a quadratic inequality must be solved together with another condition.
For example, a problem may require both:
and:
When conditions are connected by AND, we must find their intersection.
Example 11: Quadratic inequality with a linear restriction
Solve:
Step 1. Factorize the quadratic inequality.
The roots are 1 and 4.
Because the parabola opens upward, the quadratic inequality is satisfied between these roots.
Therefore:
Interval notation:
Step 2. Consider the additional condition.
Interval notation:
Step 3. Find the intersection.
We need values satisfying both conditions.
Final answer:
Notice that 2 is included because it satisfies both original conditions, while 4 is excluded because the quadratic inequality is strict.
01.5.11 Applying Quadratic Inequalities to Word Problems
Quadratic inequalities are useful for solving real-world problems involving area, dimensions, optimization constraints, and other relationships containing squared variables.
Example 12: Rectangle area problem
A rectangular study room has a width of \(x\) meters and a length that is 3 meters greater than its width.
The room must have an area of at least 40 square meters.
Find the possible values of \(x\).
Step 1. Define the dimensions.
Width:
Length:
Since these are physical dimensions:
Step 2. Form the inequality.
The area of a rectangle is:
Therefore:
The room must have an area of at least 40 square meters.
Thus:
Step 3. Expand and simplify.
Move all terms to the left:
Step 4. Factorize.
The roots are:
Step 5. Solve the quadratic inequality.
The quadratic opens upward and is nonnegative outside its roots.
Therefore:
Step 6. Apply the real-world restriction.
Because \(x\) represents the width of a room, it must be positive.
The condition \(x\leq-8\) is not physically meaningful.
Therefore:
Final answer: The width must be at least 5 meters.
At the minimum width of 5 meters, the length is 8 meters, giving an area of exactly 40 square meters.
Important principle: After solving an inequality, check whether all algebraic solutions are valid in the original problem.
01.5.12 Common Mistakes When Solving Quadratic Inequalities
Mistake 1: Giving only the roots as the answer
Consider:
Factorize:
The roots are:
These roots are boundary points, not the complete solution set.
The correct answer is:
Interval notation:
Mistake 2: Choosing the wrong intervals
Consider:
It is incorrect to conclude:
Because the parabola opens upward, the expression is positive outside the roots.
The correct solution is:
Mistake 3: Forgetting to include the roots
Consider:
Factorize:
The correct solution is:
Both −1 and 1 must be included because the inequality permits equality.
Mistake 4: Forgetting to reverse the sign when multiplying by −1
Consider:
Multiplying by −1 gives:
not:
The correct solution is:
Mistake 5: Assuming every quadratic inequality has two roots
Consider:
Since:
for all real \(x\), we have:
Therefore, the inequality has no real solution:
∅
Mistake 6: Dividing by an unknown variable expression
Consider:
It would be incorrect to divide both sides by \(x\) without checking whether \(x\) is positive, negative, or zero.
The correct method is to analyze the signs of both factors.
The solution is:
Mistake 7: Assuming the sign changes at every root
Consider:
The expression is positive on both sides of the repeated root \(x=2\).
It does not become negative when crossing that root.
The solution is:
Remember that a repeated root does not cause a quadratic expression to change sign.
01.5.13 General Strategy for Solving Quadratic Inequalities
When solving a quadratic inequality, use the following decision process:
Choosing an efficient method
| Situation | Recommended method |
|---|---|
| The quadratic factorizes easily | Factorization and sign analysis |
| The roots are irrational or difficult to identify | Quadratic formula |
| A perfect square is present | Square properties |
| The discriminant is negative | Determine the sign from \(a\) |
| A complete-square form is available | Completing the square |
| A graph is provided | Analyze above and below the x-axis |
| Additional restrictions are given | Solve, then intersect solution sets |
01.5.14 Lesson Summary
Quadratic inequalities require students to determine the sign of a quadratic expression over different intervals.
The most important points are:
-
A quadratic inequality contains a quadratic expression whose highest power is 2.
-
Always rewrite the inequality with zero on one side before analyzing its sign.
-
The roots of the related quadratic equation identify possible boundary points.
-
A quadratic with a positive leading coefficient and two distinct real roots is positive outside the roots and negative between them.
-
A quadratic with a negative leading coefficient and two distinct real roots follows the opposite pattern.
-
Strict inequalities exclude roots where the expression equals zero.
-
Non-strict inequalities include those roots.
-
A quadratic with a repeated root does not change sign at that root.
-
A quadratic with no real roots maintains a constant sign determined by its leading coefficient.
-
A sign chart, graph, or quadratic formula can be used when factorization alone is insufficient.
Essential rules to remember
For a quadratic expression with two distinct real roots \(r_1<r_2\):
If \(a>0\):
has solution:
And:
has solution:
If \(a<0\), these sign regions are reversed.
For non-strict inequalities (\(\geq,\leq\)), include both roots as appropriate.
Final reminder: Do not memorize the interval patterns without understanding their meaning. Finding the roots and determining the sign of the expression are two separate steps, and both are required to solve quadratic inequalities correctly.