What is the difference between solution sets for vs quadratic inequalities?

For a quadratic with positive coefficient: <ul<liFor : Solution values are outside the two roots, written as or </li<liFor : Solution values are between the two roots, written as </li</ul Sketch a quick upward-opening parabola to confirm.

Discriminant, Roots and Quadratic Inequalities

Q: How do I check if a line is tangent to a quadratic curve?

Substitute the line equation into the quadratic curve equation to form a single quadratic equation in one variable. If its discriminant equals 0, the line is tangent to the curve.

CIE IGCSE Additional Mathematics· 2.3, 2.4, 2.5· 25 min read

1. 1. The Discriminant and Classification of Real Roots★★☆☆☆⏱ 6 min

📘 Definition

Discriminant

\Delta = b^2 - 4ac

For a quadratic equation of the form $a x^{2} + b x + c = 0$ where $a \neq = 0$ , the discriminant is the value that determines the number of real roots of the equation.

Example:

For $2 x^{2} + 5 x + 2 = 0$ , $Δ = 5^{2} - 4 (2) (2) = 9$

The discriminant maps to three possible cases for real roots (complex roots are not assessed in 0606):

$Δ > 0$ : Two distinct real roots
$Δ = 0$ : One repeated (equal) real root
$Δ < 0$ : No real roots

📐 Worked Example

Find the values of $k$ for which the quadratic equation $3 x^{2} + k x + 12 = 0$ has equal real roots.

1
Identify coefficients: $a = 3$ , $b = k$ , $c = 12$
2
For equal roots, discriminant $Δ = 0$
3
$k^2 - 4(3)(12) = 0$
4
$k^2 - 144 = 0 \implies k = \pm 12$

2. 2. Discriminant Applications: Line and Curve Intersections★★★☆☆⏱ 7 min

When analyzing if a straight line intersects a quadratic parabola, substitute the line equation into the curve equation to form a single quadratic equation in one variable. The discriminant of this resulting equation tells you the number of intersection points:

$Δ > 0$ : Two distinct intersection points
$Δ = 0$ : One intersection point (the line is tangent to the curve)
$Δ < 0$ : No intersection points

📐 Worked Example

Find the value of $m$ for which the line $y = m x + 3$ is tangent to the curve $y = 2 x^{2} + 5 x + 4$ .

1
Equate the two expressions for $y$ :
2
$mx + 3 = 2x^2 + 5x + 4$
3
Rearrange into standard quadratic form:
4
$2x^2 + (5 - m)x + 1 = 0$
5
For tangent condition, $Δ = 0$ :
6
$(5 - m)^2 - 4(2)(1) = 0$
7
$m^2 - 10m + 17 = 0$
8
Solve using quadratic formula:
9
$m = 5 \pm 2\sqrt{2}$

Exam tip:

This is a frequent 3-4 mark exam question: always follow the sequence substitute → rearrange → apply discriminant rule.

3. 3. Solving Quadratic Inequalities★★★☆☆⏱ 7 min

Quadratic inequalities are expressions of the form $a x^{2} + b x + c > 0$ , $a x^{2} + b x + c < 0$ , $a x^{2} + b x + c \geq 0$ , or $a x^{2} + b x + c \leq 0$ . Follow these steps to solve them:

Rearrange the inequality so all terms are on one side, with the coefficient of $x^{2}$ positive.
Solve the corresponding quadratic equation to find the roots.
Sketch a quick upward-opening parabola through the roots to identify the region that satisfies the inequality.
Write the solution set using correct notation.

📐 Worked Example

Solve the inequality $2 x^{2} - 5 x - 3 < 0$ .

1
Coefficient of $x^{2}$ is already positive, so no sign change is needed.
2
Solve the corresponding quadratic equation by factorisation:
3
$(2x + 1)(x - 3) = 0 \implies \text{roots at } x = -0.5, x = 3$
4
Sketch upward opening parabola crossing x-axis at $- 0.5$ and $3$ . The region where the parabola is below the x-axis (for $< 0$ ) is between the two roots.
5
Solution set: $- 0.5 < x < 3$

4. 4. Correct Notation for Quadratic Inequality Solutions★★☆☆☆⏱ 5 min

Quadratic inequalities produce two types of solution sets for positive $x^{2}$ coefficients:

For $< 0$ or $\leq 0$ : Solution is a single interval between the two roots, written as $a < x < b$ .
For $> 0$ or $\geq 0$ : Solution is two separate intervals, written as $x < a$ or $x > b$ . Never write $a > x > b$ for two intervals, as this implies $x$ is simultaneously less than $a$ and greater than $b$ , which is impossible.

✓ Quick check

What is the solution set for $(x - 2) (x + 4) > 0$ ?
- $- 4 < x < 2$
- $x < - 4$ or $x > 2$
- $x < - 2$ or $x > 4$
Reveal answer
B —
For $> 0$ with positive leading coefficient, values are outside the roots $x = - 4$ and $x = 2$ .
What is the solution set for $x^{2} - 6 x + 9 \leq 0$ ?
Reveal answer
$x = 3$ —
The quadratic factorises to $(x - 3)^{2} = 0$ , which has a repeated root at $x = 3$ , so only $x = 3$ satisfies the inequality.

5. Common Pitfalls

Wrong move:

Forgetting to make the coefficient of $x^{2}$ positive before solving a quadratic inequality

Why:

This flips the direction of the parabola, leading to the wrong region being selected for the solution set

Correct move:

Multiply both sides of the inequality by -1, reversing the inequality sign, to make the $x^{2}$ coefficient positive before finding roots

Wrong move:

Writing a solution set for a $> 0$ inequality as $a < x < b$

Why:

This incorrectly selects the region between the roots, which is the region where the quadratic is $< 0$ for positive leading coefficient

Correct move:

For positive leading coefficient and $> 0$ inequality, select the regions outside the two roots, written as $x < a$ or $x > b$

Wrong move:

Using the discriminant for non-quadratic equations where $a = 0$

Why:

The discriminant formula only applies to equations of the form $a x^{2} + b x + c = 0$ where $a \neq = 0$

Correct move:

First confirm the coefficient of $x^{2}$ is non-zero before applying discriminant rules

Wrong move:

Rounding discriminant values too early when solving line-curve intersection problems

Why:

This leads to incorrect root values, losing accuracy marks

Correct move:

Keep values in exact surd form until the final step, unless told otherwise by the question

6. Quick Reference Cheatsheet

Concept	Condition	Result
Discriminant for $a x^{2} + b x + c = 0$	$Δ > 0$	Two distinct real roots
	$Δ = 0$	One repeated real root
	$Δ < 0$	No real roots
Line-curve intersection	$Δ > 0$	Two intersection points
	$Δ = 0$	Line is tangent to curve
	$Δ < 0$	No intersection points
Quadratic inequality ( $a > 0$ )	$a x^{2} + b x + c < 0$	Solution: between roots ( $a < x < b$ )
	$a x^{2} + b x + c > 0$	Solution: outside roots ( $x < a$ or $x > b$ )

7. Frequently Asked

What is the difference between solution sets for $> 0$ vs $< 0$ quadratic inequalities?

For a quadratic with positive $x^{2}$ coefficient:

For $> 0$ : Solution values are outside the two roots, written as $x < a$ or $x > b$
For $< 0$ : Solution values are between the two roots, written as $a < x < b$

Sketch a quick upward-opening parabola to confirm.

How do I check if a line is tangent to a quadratic curve?

Substitute the line equation into the quadratic curve equation to form a single quadratic equation in one variable. If its discriminant equals 0, the line is tangent to the curve.

What's Next

Now that you have mastered discriminant analysis and quadratic inequalities, you are ready to apply these skills to more advanced quadratic function topics in CIE IGCSE Additional Mathematics 0606. These concepts are foundational for upcoming units including quadratic graph sketching, maximum and minimum value problems, and kinematics applications where quadratic motion equations are used. You will also encounter discriminant rules again when solving simultaneous equations involving one linear and one quadratic equation, a common 5-6 mark exam question. Practice these skills regularly using past paper questions to build speed and avoid common sign errors, especially when rearranging inequalities and applying discriminant conditions.