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 \(D\) or \(\Delta\)) is the quantity sitting under the square-root sign in the quadratic formula:

\(D = b^{2} - 4ac\)

Because the roots are \(x = \dfrac{-b \pm \sqrt{D}}{2a}\), the sign of \(D\) alone tells us how many real roots there are and what kind they are — without solving the equation. This is why \(D\) is called the discriminant: it "discriminates" between the cases.

Nature of roots based on the discriminant:

DiscriminantNature of rootsGeometric meaning
\(D > 0\)Two distinct real rootsParabola cuts the x-axis at two points
\(D = 0\)Two equal real roots (one repeated root)Parabola just touches the x-axis
\(D < 0\)No real roots (complex roots)Parabola never meets the x-axis

Why this works. When \(D>0\), \(\sqrt{D}\) is a positive real number, so \(\dfrac{-b+\sqrt{D}}{2a}\) and \(\dfrac{-b-\sqrt{D}}{2a}\) are two different reals. When \(D=0\), \(\sqrt{D}=0\) and both formulas collapse to the single value \(x=-\dfrac{b}{2a}\) — a repeated root. When \(D<0\), \(\sqrt{D}\) is not a real number, so there are no real roots (the roots are a pair of complex conjugates, studied later).

A special case of \(D>0\): perfect-square discriminant. If \(D\) is a perfect square (and \(a,b,c\) are rational), the roots are rational and the equation can be factorised over the rationals. If \(D>0\) but is not a perfect square, the roots are real but irrational (they involve a surd). This is a handy quick test before attempting factorisation.

The quadratic formula in full. When \(D \geq 0\) the roots are real and given by \(x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}\). This is the most reliable method for messy coefficients and for any speed/time, work-rate, or tank-filling word problem that does not factor neatly. After computing \(D\), if it is non-negative, substitute straight into the formula.

Finding an unknown for a required nature of roots. A very common exam task is: "find the value(s) of \(k\) for which the equation has equal roots." The recipe is mechanical:

  • For equal roots, set \(D = 0\) and solve for \(k\).
  • For two distinct real roots, require \(D > 0\) (an inequality in \(k\)).
  • For no real roots, require \(D < 0\).
  • For real roots (distinct or equal), require \(D \geq 0\).

Take care to identify \(a, b, c\) correctly first — many sign errors creep in when \(b\) itself contains \(k\), as in \(kx^{2}-2x+1=0\) or \(2x^{2}+kx+3=0\).

Sum and product of roots (useful relations). If \(\alpha\) and \(\beta\) are the roots of \(ax^{2}+bx+c=0\), then \(\alpha+\beta = -\dfrac{b}{a}\) and \(\alpha\beta = \dfrac{c}{a}\). These let you answer "one root is twice the other" or "roots are reciprocals" quickly without full solving.

Common mistakes to avoid: writing the discriminant as \(b^{2}+4ac\) (the sign is minus); concluding "no solution" when \(D=0\) (there is a repeated real root); forgetting that \(k\) may appear in \(b\) so that \(b^{2}\) involves \(k^{2}\); and dropping the second value of \(k\) when \(D=0\) gives a quadratic in \(k\) (e.g. \(k=\pm8\)).

Three parabolas showing how the discriminant decides the nature of the roots Discriminant D = b^2 − 4ac decides the roots D > 0 two distinct roots D = 0 one repeated root D < 0 no real roots The parabola y = ax^2 + bx + c cuts, touches, or misses the x-axis. Flowchart deciding the nature of roots from the value of the discriminant Decision flow from D compute D = b^2 − 4ac D > 0 D = 0 D < 0 two distinct real roots one repeated real root no real roots When D ≥ 0, roots are x = (−b ± √D) / 2a.
Explore it yourself
Solved Examples
1
Worked Example
Find the discriminant of \(x^{2} - 7x + 10 = 0\) and state the nature of its roots.
Solution
  1. Step 1: Compare: \(a = 1,\ b = -7,\ c = 10\).
  2. Step 2: \(D = b^{2} - 4ac = (-7)^{2} - 4(1)(10) = 49 - 40 = 9\).
  3. Step 3: \(D = 9 > 0\), and 9 is a perfect square, so the roots are real, distinct, and rational.

Answer: \(D = 9\); two distinct real (rational) roots

2
Worked Example
Find the discriminant of \(2x^{2} - 6x + 3 = 0\) and state the nature of its roots.
Solution
  1. Step 1: \(a = 2,\ b = -6,\ c = 3\).
  2. Step 2: \(D = (-6)^{2} - 4(2)(3) = 36 - 24 = 12\).
  3. Step 3: \(D = 12 > 0\) but 12 is not a perfect square, so the roots are real, distinct, and irrational.

Answer: \(D = 12\); two distinct real (irrational) roots

3
Worked Example
Examine the nature of the roots of \(x^{2} + x + 1 = 0\).
Solution
  1. Step 1: \(a = 1,\ b = 1,\ c = 1\).
  2. Step 2: \(D = (1)^{2} - 4(1)(1) = 1 - 4 = -3\).
  3. Step 3: \(D = -3 < 0\), so there are no real roots.

Answer: \(D = -3\); no real roots

4
Worked Example
Show that \(x^{2} - 4x + 4 = 0\) has equal roots and find the repeated root.
Solution
  1. Step 1: \(a = 1,\ b = -4,\ c = 4\).
  2. Step 2: \(D = (-4)^{2} - 4(1)(4) = 16 - 16 = 0\).
  3. Step 3: Since \(D = 0\), the roots are real and equal.
  4. Step 4: The repeated root is \(x = -\dfrac{b}{2a} = -\dfrac{-4}{2} = 2\).

Answer: \(D = 0\); equal roots, \(x = 2\)

5
Worked Example
For what value of \(k\) does \(4x^{2} - 12x + k = 0\) have equal roots?
Solution
  1. Step 1: For equal roots, set \(D = 0\).
  2. Step 2: \(a = 4,\ b = -12,\ c = k\).
  3. Step 3: \(D = (-12)^{2} - 4(4)(k) = 144 - 16k\).
  4. Step 4: Set \(144 - 16k = 0 \Rightarrow 16k = 144 \Rightarrow k = 9\).

Answer: \(k = 9\)

6
Worked Example
Find the values of \(k\) for which \(2x^{2} + kx + 8 = 0\) has equal roots.
Solution
  1. Step 1: Equal roots require \(D = 0\).
  2. Step 2: \(a = 2,\ b = k,\ c = 8\).
  3. Step 3: \(D = k^{2} - 4(2)(8) = k^{2} - 64\).
  4. Step 4: Set \(k^{2} - 64 = 0 \Rightarrow k^{2} = 64 \Rightarrow k = \pm 8\).
  5. Step 5: Do not drop the negative value — both satisfy the condition.

Answer: \(k = 8\) or \(k = -8\)

7
Worked Example
For what values of \(k\) does \(kx^{2} - 2x + 1 = 0\) have real roots? (Assume \(k \neq 0\).)
Solution
  1. Step 1: Real roots require \(D \geq 0\).
  2. Step 2: \(a = k,\ b = -2,\ c = 1\).
  3. Step 3: \(D = (-2)^{2} - 4(k)(1) = 4 - 4k\).
  4. Step 4: Require \(4 - 4k \geq 0 \Rightarrow 4 \geq 4k \Rightarrow k \leq 1\).
  5. Step 5: Combined with \(k \neq 0\): real roots for \(k \leq 1,\ k \neq 0\).

Answer: \(k \leq 1\) (and \(k \neq 0\))

8
Worked Example
Find the roots of \(2x^{2} - 4x + 1 = 0\) using the quadratic formula.
Solution
  1. Step 1: \(a = 2,\ b = -4,\ c = 1\).
  2. Step 2: \(D = (-4)^{2} - 4(2)(1) = 16 - 8 = 8 > 0\), so two distinct real roots.
  3. Step 3: \(\sqrt{D} = \sqrt{8} = 2\sqrt{2}\).
  4. Step 4: \(x = \dfrac{-b \pm \sqrt{D}}{2a} = \dfrac{4 \pm 2\sqrt{2}}{4}\).
  5. Step 5: Simplify: \(x = \dfrac{2 \pm \sqrt{2}}{2}\).

Answer: \(x = \dfrac{2 + \sqrt{2}}{2}\) and \(x = \dfrac{2 - \sqrt{2}}{2}\)

9
Worked Example
Solve \(x^{2} + 6x + 6 = 0\) using the quadratic formula.
Solution
  1. Step 1: \(a = 1,\ b = 6,\ c = 6\).
  2. Step 2: \(D = 6^{2} - 4(1)(6) = 36 - 24 = 12 > 0\).
  3. Step 3: \(\sqrt{12} = 2\sqrt{3}\).
  4. Step 4: \(x = \dfrac{-6 \pm 2\sqrt{3}}{2} = -3 \pm \sqrt{3}\).

Answer: \(x = -3 + \sqrt{3},\ x = -3 - \sqrt{3}\)

10
Worked Example
If the roots of \((b - c)x^{2} + (c - a)x + (a - b) = 0\) are equal, prove that \(2b = a + c\).
Solution
  1. Step 1: Equal roots mean \(D = 0\), where \(A = b-c,\ B = c-a,\ C = a-b\).
  2. Step 2: \(D = (c - a)^{2} - 4(b - c)(a - b) = 0\).
  3. Step 3: Expand: \((c-a)^{2} = c^{2} - 2ac + a^{2}\), and \(4(b-c)(a-b) = 4(ab - b^{2} - ac + bc)\).
  4. Step 4: So \(c^{2} - 2ac + a^{2} - 4ab + 4b^{2} + 4ac - 4bc = 0\), i.e. \(a^{2} + 4b^{2} + c^{2} + 2ac - 4ab - 4bc = 0\).
  5. Step 5: This is exactly \((a + c - 2b)^{2} = 0\) (a perfect square).
  6. Step 6: Hence \(a + c - 2b = 0\), that is \(2b = a + c\).

Answer: \(2b = a + c\) (proved)

11
Worked Example
The sum of a number and its reciprocal is \(\dfrac{10}{3}\). Find the number using a quadratic equation.
Solution
  1. Step 1: Let the number be \(x\); then \(x + \dfrac{1}{x} = \dfrac{10}{3}\).
  2. Step 2: Multiply by \(3x\): \(3x^{2} + 3 = 10x\).
  3. Step 3: Standard form: \(3x^{2} - 10x + 3 = 0\).
  4. Step 4: \(D = (-10)^{2} - 4(3)(3) = 100 - 36 = 64\); \(\sqrt{64} = 8\).
  5. Step 5: \(x = \dfrac{10 \pm 8}{6}\), giving \(x = 3\) or \(x = \dfrac{1}{3}\).

Answer: The number is 3 (or its reciprocal \(\dfrac{1}{3}\))

12
Worked Example
A motor boat whose speed in still water is 18 km/h takes 1 hour more to go 24 km upstream than to return downstream to the same spot. Find the speed of the stream.
Solution
  1. Step 1: Let the stream speed be \(x\) km/h. Upstream speed \(= 18 - x\), downstream speed \(= 18 + x\).
  2. Step 2: Upstream time \(= \dfrac{24}{18 - x}\), downstream time \(= \dfrac{24}{18 + x}\); upstream is 1 hour more.
  3. Step 3: \(\dfrac{24}{18 - x} - \dfrac{24}{18 + x} = 1\).
  4. Step 4: Combine: \(\dfrac{24(18 + x) - 24(18 - x)}{(18 - x)(18 + x)} = 1\Rightarrow \dfrac{48x}{324 - x^{2}} = 1\).
  5. Step 5: Cross-multiply: \(48x = 324 - x^{2}\Rightarrow x^{2} + 48x - 324 = 0\).
  6. Step 6: \(D = 48^{2} + 4(324) = 2304 + 1296 = 3600\); \(\sqrt{3600} = 60\).
  7. Step 7: \(x = \dfrac{-48 \pm 60}{2}\), giving \(x = 6\) or \(x = -54\). Reject the negative speed.

Answer: Speed of the stream = 6 km/h

Key Points

  • Discriminant \(D = b^{2} - 4ac\) (note the minus sign) decides the nature of the roots
  • \(D > 0\) → two distinct real roots; if \(D\) is a perfect square the roots are rational, else irrational
  • \(D = 0\) → two equal real roots (a repeated root \(x = -\dfrac{b}{2a}\)) — this still counts as having real roots
  • \(D < 0\) → no real roots (the roots are complex conjugates)
  • Quadratic formula: \(x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}\), valid whenever \(D \geq 0\)
  • For equal roots set \(D = 0\); for real roots set \(D \geq 0\); for no real roots set \(D < 0\)
  • Sum of roots \(= -\dfrac{b}{a}\), product of roots \(= \dfrac{c}{a}\) — useful for ratio/reciprocal questions
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.
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