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Fundamentals of Business Mathematics and Statistics · Quadratic Equations

Nature of Roots of a Quadratic Equation Using the Discriminant

Updated 10 October 2026 · Fact-checked

The discriminant D = b² − 4ac of the quadratic ax² + bx + c = 0 (a ≠ 0) tells you the nature of its roots without solving. If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, roots are imaginary (not real).

Understand Nature of Roots and Discriminant

A quadratic equation ax² + bx + c = 0 has two roots. You find them with the formula x = [−b ± √(b² − 4ac)] ÷ 2a. Look at the part under the square root. That part decides what kind of roots you get.

That part is called the discriminant, written D. So D = b² − 4ac. It "discriminates", or separates, the three possible cases. You only need to find D, not the roots.

If D is positive, √D is a real number and the ± gives two different values. The roots are real and distinct. If D is zero, √D = 0, so both values are the same: x = −b ÷ 2a. The roots are real and equal. If D is negative, √D is not a real number. The roots are imaginary (complex), and the graph never touches the x-axis.

There is one more useful point. If a, b, c are rational numbers and D is a perfect square of a rational number, the roots are rational. If D is positive but not a perfect square, the roots are irrational. Also, when the coefficients are rational, irrational roots come in pairs like p + √q and p − √q.

The same idea works in reverse. If a question says roots are equal, put D = 0. If it says roots are real, put D ≥ 0. Then solve for the unknown constant, often k.

Key formulas to remember

Discriminant
D = b² − 4ac
For ax² + bx + c = 0 with a ≠ 0. Always write the equation in standard form first.
Real and distinct roots
D > 0
Two different real roots. If D is also a perfect square (rational coefficients), the roots are rational.
Real and equal roots
D = 0
Each root is −b ÷ 2a. The condition is b² = 4ac.
Imaginary roots
D < 0
No real roots. Roots are complex conjugates.
Real roots (either type)
D ≥ 0
Use when a question says only 'roots are real'.
Roots by formula
x = [−b ± √D] ÷ 2a
The discriminant sits under the square root.

How to solve Nature of Roots and Discriminant questions

Use this method for any question on nature of roots or on finding a constant from a condition on the roots.

  1. 1Bring the equation to the form ax² + bx + c = 0 and read off a, b and c with their signs.
  2. 2Write D = b² − 4ac. If a constant such as k is present, keep it in the expression.
  3. 3Substitute the values carefully. Square b first, then subtract 4ac.
  4. 4For a numerical equation, compare D with zero and state the nature: D > 0, D = 0 or D < 0.
  5. 5If D > 0 and the options mention rational or irrational, check whether D is a perfect square.
  6. 6For a condition question, set D = 0 (equal roots), D > 0 (distinct real roots) or D < 0 (imaginary roots), then solve for the constant.
  7. 7Check your answer by putting the value back into D.

Quickest way: Compute D and match the option

When to use it: When the question gives a numerical equation and asks the nature of roots, or gives k and says roots are equal.

  1. Read a, b, c directly. Do not expand or solve the equation.
  2. Compute b² and 4ac separately, then subtract.
  3. Only the sign of D matters. Stop as soon as you know it.
  4. For 'equal roots' questions, write b² = 4ac and solve for k at once.
  5. If k appears squared, expect two values of k and look for the option with both.

Common mistakes in Nature of Roots and Discriminant

  • Taking the wrong signs of a, b or c

    Students read the numbers without the minus signs, or forget that a missing term means 0.

    Fix: Write a = , b = , c = with signs before using the formula. If x is missing, b = 0.

  • Writing D = b² − 4ac as b² − 4 + ac or (b − 4ac)²

    Rushing and misremembering the formula.

    Fix: Remember it as 'b squared minus four a c'. Compute b² and 4ac separately.

  • Mishandling −4ac when c or a is negative

    Subtracting a negative product is easy to get wrong. For a = 2, c = −3, −4ac = +24.

    Fix: Find 4ac with its sign first, then subtract it.

  • Not rearranging to standard form

    The equation is given as 2x² = 3x − 5 or x(x + 2) = 4, and students use the numbers as they appear.

    Fix: Move everything to one side so the right side is 0 before reading a, b and c.

  • Using D > 0 when the question says 'real roots'

    Real roots includes the equal case.

    Fix: 'Real' means D ≥ 0. 'Real and distinct' means D > 0.

  • Giving only one value of k

    Setting D = 0 often gives a quadratic in k with two solutions.

    Fix: Solve the equation in k fully and check which options match.

Worked examples

Example 1

The nature of the roots of 2x² − 5x + 4 = 0 is: (a) real and distinct (b) real and equal (c) imaginary (d) rational and distinct

Show the solution
  1. Here a = 2, b = −5, c = 4.
  2. D = b² − 4ac = (−5)² − 4 × 2 × 4.
  3. D = 25 − 32 = −7.
  4. D < 0, so the roots are not real.

Answer: (c) imaginary

Example 2

For what value of k does x² + kx + 25 = 0 have equal roots, given k > 0? (a) 5 (b) 10 (c) 20 (d) 25

Show the solution
  1. Here a = 1, b = k, c = 25.
  2. Equal roots need D = 0, so k² − 4 × 1 × 25 = 0.
  3. k² = 100, so k = 10 or k = −10.
  4. The question says k > 0, so k = 10.
  5. Check: x² + 10x + 25 = (x + 5)², which has equal roots x = −5.

Answer: (b) 10

Exam tips

  • Most questions are one-step: compute D and read its sign. Aim to finish these in under a minute.
  • In 'find k' questions, convert the words into a condition: equal means D = 0, real means D ≥ 0, imaginary means D < 0.
  • If the options include both rational and irrational, test whether D is a perfect square before choosing.
  • With no negative marking, always attempt every question. For a constant k, put an option value into D to test it.

Practice questions from Quadratic Equations

Nature of Roots and Discriminant: frequently asked questions

What is the discriminant of a quadratic equation?

It is the expression b² − 4ac for the equation ax² + bx + c = 0. It is the quantity under the square root in the quadratic formula. Its sign tells you the nature of the roots.

What is the condition for equal roots?

The roots are real and equal when D = 0, that is b² = 4ac. Each root is then −b ÷ 2a. This condition is used to find unknown constants like k.

Can a quadratic equation have only one root?

When D = 0, the two roots coincide, so you get one repeated value. It is still counted as two equal roots. When D < 0 the roots are imaginary, not missing.

How do I tell if the roots are rational?

For an equation with rational coefficients, find D. If D is positive and a perfect square, the roots are rational and distinct. If D is positive but not a perfect square, the roots are irrational.