Quadratic Equations • Topic 3 of 3

Discriminant and Nature of Roots

What is the discriminant? For a quadratic equation $ax^{2} + bx + c$ = 0, the discriminant (denoted by D or Δ) is: D = $b^{2} - 4ac$

The discriminant tells us the nature of the roots without actually solving the equation!

Nature of roots based on discriminant:

Discriminant (D)	Nature of Roots	Roots are
D > 0	Real and distinct	Two different real numbers
D = 0	Real and equal (repeated)	One real number (double root)
D < 0	No real roots (imaginary)	Two complex conjugate roots

Quadratic formula: When D ≥ 0, the roots are given by: x = [−b ± $\sqrt{D}$] / (2a)

Real-life meaning:

D > 0 → The parabola (graph of quadratic) cuts the x-axis at two different points
D = 0 → The parabola touches the x-axis at exactly one point (vertex on axis)
D < 0 → The parabola never touches the x-axis (lies entirely above or below)

┌─────────────────────────────────────────────────────────────┐
│         DISCRIMINANT AND NATURE OF ROOTS - GRAPH MAP         │
└─────────────────────────────────────────────────────────────┘

CASE 1: D > 0 (Two distinct real roots)

        y
        │   \     /
        │    \   /
        │     \ /
        │      X
        │     / \
        │    /   \
        │   /     \
        └───┼───┼───► x
            r1  r2
    
    Example: x² - 5x + 6 = 0 → D = 25-24=1>0, roots: 2,3


CASE 2: D = 0 (One repeated real root)

        y
        │     ┌┐
        │    ╱  ╲
        │   ╱    ╲
        │  ╱      ╲
        │ ╱        ╲
        │╱          ╲
        └─────┼──────► x
              r
    
    Example: x² - 4x + 4 = 0 → D = 16-16=0, root: 2 (double)


CASE 3: D < 0 (No real roots)

        y
        │     ┌┐
        │    ╱  ╲
        │   ╱    ╲
        │  ╱      ╲
        │ ╱        ╲
        │╱          ╲
        └──────────────► x
        (never touches x-axis)
    
    Example: x² + x + 1 = 0 → D = 1-4 = -3 < 0, no real roots


DISCRIMINANT FLOWCHART:

        Start: ax² + bx + c = 0
                │
                ▼
        Calculate D = b² - 4ac
                │
        ┌───────┼───────┐
        ▼       ▼       ▼
       D>0     D=0     D<0
        │       │       │
        ▼       ▼       ▼
    Two real  One real  No real
    distinct  root      roots
      roots    (double)  
        │       │
        └───┬───┘
            ▼
    Use quadratic formula: x = (-b ± √D)/(2a)

Solved Examples

Worked Example

Find the discriminant of $x^{2} - 7x + 10$ = 0 and state the nature of roots.

Solution

Step 1: Compare with $ax^{2} + bx + c$ = 0 → a = 1, b = −7, c = 10
Step 2: D = $b^{2} - 4ac$ = (−7)$^{2} - 4$(1)(10) = 49 − 40 = 9
Step 3: Since D = 9 > 0, roots are real and distinct

Answer: D = 9; roots are real and distinct

Worked Example

For what value of k does $4x^{2} - 12x + k$ = 0 have equal roots?

Solution

Step 1: For equal roots, discriminant D = 0
Step 2: a = 4, b = −12, c = k
Step 3: D = (−12)$^{2} - 4$(4)(k) = 144 − 16k
Step 4: Set D = 0: 144 − 16k = 0 → 16k = 144 → k = 9

Answer: k = 9

Worked Example

Find the roots of $2x^{2} - 4x + 1$ = 0 using the quadratic formula.

Solution

Step 1: a = 2, b = −4, c = 1
Step 2: D = $b^{2} - 4ac$ = (−4)$^{2} - 4$(2)(1) = 16 − 8 = 8
Step 3: Since D > 0, roots are real and distinct
Step 4: $\sqrt{D}$ = $\sqrt{8}$ = 2$\sqrt{2}$
Step 5: x = [−b ± $\sqrt{D}$]/(2a) = [4 ± 2$\sqrt{2}$]/(4)
Step 6: Simplify: x = [2(2 ± $\sqrt{2}$)]/4 = (2 ± $\sqrt{2}$)/2

Answer: x = (2 + $\sqrt{2}$)/2 and x = (2 − $\sqrt{2}$)/2

Key Points

Discriminant D = $b^{2} - 4ac$ determines the nature of roots
D > 0 → two distinct real roots
D = 0 → one real repeated root (double root)
D < 0 → no real roots (complex roots)
Quadratic formula: x = [−b ± $\sqrt{\(b^{2} - 4ac$}\)]/(2a)
The quadratic formula works for any quadratic equation

Quick Check — 4 MCQs

Tap an option to check your answer0 / 4

Q1.The discriminant of $ax^2+bx+c=0$ is:

Explanation: $D=b^2-4ac$.

Q2.If $D>0$, the roots are:

Explanation: Two distinct real roots.

Q3.If $D=0$, the roots are:

Explanation: Repeated real root.

Q4.If $D<0$, the roots are:

Explanation: No real roots.

Practice more MCQs → 📝 Assignment · 35 marks