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Class 10 — Mathematics (NCERT)

Quadratic Equations

Class 10

  • ✓Define a quadratic equation and write it in its standard form.
  • ✓Solve quadratic equations by the method of factorisation (splitting the middle term).
  • ✓Solve quadratic equations using the quadratic formula.
  • ✓Determine the nature of roots of a quadratic equation using the discriminant.

Key concepts

Introduction to Quadratic Equation

An equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0, is called a quadratic equation. Here, 'x' is the variable and 'a', 'b', 'c' are coefficients. The highest power of the variable in a quadratic equation is 2.

ax² + bx + c = 0
Solving Quadratic Equations by Factorisation

This method involves expressing the quadratic polynomial ax² + bx + c as a product of two linear factors. We achieve this by splitting the middle term 'bx' into two terms such that their sum is 'bx' and their product is 'acx²'. After splitting, we factorise by grouping terms. If the equation can be written as (x - α)(x - β) = 0, then the roots are x = α and x = β.

Solving Quadratic Equations by Quadratic Formula

For a quadratic equation in standard form ax² + bx + c = 0 (where a ≠ 0), the roots are given by the quadratic formula. This formula provides a direct method to find the roots and is particularly useful when factorisation is difficult or not possible.

x = [-b ± sqrt(b² - 4ac)] / 2a
Nature of Roots

The nature of the roots of a quadratic equation ax² + bx + c = 0 is determined by the value of the discriminant, D = b² - 4ac. The discriminant tells us whether the roots are real and distinct, real and equal, or not real.

D = b² - 4ac

Key facts to remember

  • 1The standard form of a quadratic equation is ax² + bx + c = 0, where a ≠ 0.
  • 2A quadratic equation has at most two roots (solutions).
  • 3The quadratic formula to find the roots of ax² + bx + c = 0 is x = [-b ± sqrt(b² - 4ac)] / 2a.
  • 4The discriminant (D) of a quadratic equation is given by D = b² - 4ac.
  • 5If D > 0, the quadratic equation has two distinct real roots.
  • 6If D = 0, the quadratic equation has two equal real roots (or coincident roots).
  • 7If D < 0, the quadratic equation has no real roots (the roots are complex, which is not in Class 10 syllabus).

Worked examples

Example 1

Solve the quadratic equation 6x² - x - 2 = 0 by factorisation.

IGiven equation is 6x² - x - 2 = 0.
IIHere, a = 6, b = -1, c = -2.
IIIWe need to find two numbers whose sum is -1 and whose product is (6)(-2) = -12. These numbers are -4 and 3.
IVSplit the middle term: 6x² - 4x + 3x - 2 = 0.
VGroup the terms: 2x(3x - 2) + 1(3x - 2) = 0.
VIFactor out the common term: (3x - 2)(2x + 1) = 0.
VIISet each factor to zero: 3x - 2 = 0 or 2x + 1 = 0.
VIIISolve for x: 3x = 2 ⇒ x = 2/3 or 2x = -1 ⇒ x = -1/2.

Answer

The roots are x = 2/3 and x = -1/2.

Example 2

Find the roots of the quadratic equation 2x² + 5x - 3 = 0 using the quadratic formula.

IGiven equation is 2x² + 5x - 3 = 0.
IICompare with the standard form ax² + bx + c = 0, we have a = 2, b = 5, c = -3.
IIIThe quadratic formula is x = [-b ± sqrt(b² - 4ac)] / 2a.
IVSubstitute the values of a, b, c into the formula: x = [-(5) ± sqrt((5)² - 4(2)(-3))] / (2(2)).
VSimplify the expression under the square root: x = [-5 ± sqrt(25 + 24)] / 4.
VIx = [-5 ± sqrt(49)] / 4.
VIIx = [-5 ± 7] / 4.
VIIITwo possible values for x are:
9x₁ = (-5 + 7) / 4 = 2 / 4 = 1/2.
10x₂ = (-5 - 7) / 4 = -12 / 4 = -3.

Answer

The roots are x = 1/2 and x = -3.

Always simplify the square root and the final fractions.

Example 3

Determine the nature of roots for the equation 3x² - 2x + 1/3 = 0. If real roots exist, find them.

IGiven equation is 3x² - 2x + 1/3 = 0.
IITo work with integers, multiply the entire equation by 3: 9x² - 6x + 1 = 0.
IIICompare with ax² + bx + c = 0, we have a = 9, b = -6, c = 1.
IVCalculate the discriminant D = b² - 4ac.
VSubstitute the values: D = (-6)² - 4(9)(1).
VISimplify: D = 36 - 36.
VIID = 0.
VIIISince D = 0, the equation has two equal real roots.
9To find the roots, use the quadratic formula: x = [-b ± sqrt(D)] / 2a.
10x = [-(-6) ± sqrt(0)] / (2(9)).
11x = [6 ± 0] / 18.
12x = 6 / 18 = 1/3.

Answer

The roots are real and equal, and both are x = 1/3.

This equation is a perfect square: (3x - 1)² = 0.

Common mistakes

  • ✗Incorrectly identifying the coefficients a, b, and c, especially when terms are rearranged or missing (e.g., in x² - 4 = 0, b=0).
  • ✗Making arithmetic errors, particularly with negative signs, when calculating the discriminant or using the quadratic formula.
  • ✗Forgetting to check the value of the discriminant before attempting to find real roots, leading to attempts to find the square root of a negative number.
  • ✗Not simplifying the roots to their simplest fractional or radical form.
  • ✗Errors in splitting the middle term during factorisation, leading to incorrect factors.

Exam tips

  • ★Always write the quadratic equation in its standard form (ax² + bx + c = 0) before identifying the values of a, b, and c.
  • ★When using the quadratic formula, it is a good practice to calculate the discriminant (D = b² - 4ac) separately first. This helps in determining the nature of roots and simplifies the subsequent calculation.
  • ★Show all steps clearly in your working, especially when factorising or substituting values into the formula. This helps in avoiding calculation errors and allows for partial marks in case of a final error.
  • ★Verify your answers by substituting the obtained roots back into the original equation to ensure they satisfy it.
  • ★Practice both factorisation and the quadratic formula extensively, as some problems might be more efficiently solved by one method over the other.

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