Quadratic Equation Class 10 ICSE — Bansal Chapter 5
This page explains the quadratic equation class 10 ICSE chapter from Bansal’s Class 10 Mathematics, Chapter 5: Quadratic Equations, section by section. You will learn the standard form, the discriminant, factorisation, the quadratic formula, and substitution methods. Each section includes a worked pattern with fresh numbers so you can practise the method. The chapter PDF sits below for reference and download.
ICSE Class 10 Mathematics Bansal Chapter 5 Quadratic Equations PDF
What is inside Chapter 5
15 pages · 5 sections · 12 worked examples · 66 exercise questions
| Section | Title |
|---|---|
| 5.1 | Introduction |
| 5.2 | To examine the nature of the roots |
| 5.3 | Solving quadratic equations by factorisation |
| 5.4 | Solving quadratic equations using the formula |
| 5.5 | Equations Reducible to Quadratic Equations |
- EXERCISE 5(A): 5 questions
- EXERCISE 5(B): 31 questions
- EXERCISE 5(C): 10 questions
- EXERCISE 5(D): 20 questions
5.1 Introduction
A quadratic equation is an equation in one variable whose highest power is two. The standard form is ax^2 + bx + c = 0, where a, b and c are real numbers and a \neq 0.
Every quadratic equation gives two values of the unknown, called the roots. The expression b^2 - 4ac is the discriminant, written as D. It decides the nature of the roots without solving the equation.
An adfected quadratic has a square term and a first-power term, like 4x^2 + 5x = 0. A pure quadratic has only the square term, like 3x^2 - 8 = 0.
The condition a \neq 0 matters. If a were zero, the highest power would drop to one and the equation would become linear.
In 6x^2 - 11x + 5 = 0, identify a = 6, b = -11, c = 5. Classify 4x^2 - 7x = 0 as adfected and 3x^2 - 12 = 0 as pure.
5.2 To examine the nature of the roots
Examining the roots means knowing their type. Are they real or imaginary? Equal or unequal? The discriminant alone answers this.
| Discriminant | Nature of roots |
|---|---|
| D = 0 | real and equal |
| D \gt 0 | real and unequal |
| D \lt 0 | imaginary (not real) |
Every rational and irrational number is real. The square root of a negative number, like \sqrt{-4}, is imaginary. Remember the pattern: zero equal, positive unequal, negative imaginary.
For 3x^2 - 4x + 5 = 0: D = (-4)^2 - 4 \times 3 \times 5 = 16 - 60 = -44. Since D \lt 0, the roots are imaginary.
For x^2 - 10x + k = 0 to have equal roots, set D = 0. Then 100 - 4k = 0, giving k = 25.
5.3 Solving quadratic equations by factorisation
Follow four steps in order:
- Clear all fractions and brackets.
- Transpose all terms to the left to get ax^2 + bx + c = 0.
- Factorise the left side.
- Set each factor equal to zero and solve.
The zero product rule states: if the product of two expressions is zero, at least one expression must be zero. If (x + 3)(x - 2) = 0, then x + 3 = 0 or x - 2 = 0.
Never divide both sides by the variable. Dividing by x loses the root x = 0.
Solve 6x^2 - 5x - 6 = 0. Factorise: (3x + 2)(2x - 3) = 0. So 3x + 2 = 0 or 2x - 3 = 0, giving x = -\frac{2}{3} or x = \frac{3}{2}.
Build an equation from solution set \{-1, 4\}. The roots give (x + 1)(x - 4) = 0, expanding to x^2 - 3x - 4 = 0.
Test whether x = 2 is a root of 2x^2 - 5x + 2 = 0. Substitute: 2(2)^2 - 5(2) + 2 = 8 - 10 + 2 = 0. The statement holds, so x = 2 is a root.
5.4 Solving quadratic equations using the formula
For ax^2 + bx + c = 0 with a \neq 0, the roots come from the quadratic formula:
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
The proof multiplies every term by 4a, then completes the square. This turns the equation into (2ax + b)^2 = b^2 - 4ac. Taking square roots and solving for x gives the formula.
Remember it as: negative b, plus or minus the square root of b^2 minus 4ac, all over 2a.
| Mistake | Correct approach |
|---|---|
| Dropping the \pm sign | Keep \pm; it gives both roots |
| Substituting a negative b or c wrongly | Keep the sign of each coefficient intact inside the formula |
| Rounding before the final step | Round only the final answer |
Solve 2x^2 + 3x - 7 = 0. Here a = 2, b = 3, c = -7. The discriminant is D = 3^2 - 4 \times 2 \times (-7) = 9 + 56 = 65. So x = \frac{-3 \pm \sqrt{65}}{4}, giving x \approx 1.27 or x \approx -2.77, correct to 2 decimal places.
5.5 Equations Reducible to Quadratic Equations
Some equations are not quadratic at first glance. Substitution turns them into quadratics. Replace the repeated block with y, solve the quadratic in y, then substitute back to find x.
Three patterns appear often. An equation like x^4 - 5x^2 + 4 = 0 uses y = x^2. A repeated expression such as x^2 + 3x can be set as y. A reciprocal form x + \frac{1}{x} also reduces to a quadratic.
Solve x^4 - 5x^2 + 4 = 0. Put y = x^2, so the equation becomes y^2 - 5y + 4 = 0. Factorise: (y - 1)(y - 4) = 0, giving y = 1 or y = 4. Substitute back: x = \pm 1 or x = \pm 2.
Solve x + \frac{1}{x} = \frac{17}{4}. Multiply through by 4x: 4x^2 - 17x + 4 = 0. Factorise: (4x - 1)(x - 4) = 0, giving x = \frac{1}{4} or x = 4.
What to practise from this chapter
Work through the four exercise blocks in order. They build the skills in the same sequence as the chapter.
- EXERCISE 5(A) — 5 questions on the nature of roots and equal-root conditions.
- EXERCISE 5(B) — 31 questions on factorisation, root checks, and constructing equations from solution sets.
- EXERCISE 5(C) — 10 questions on the formula, rounding, and substitution.
- EXERCISE 5(D) — 20 mixed questions including parameter problems.
The textbook tags several questions with past board years. Exercise 5(A) and 5(C) carry the tags 2011, 2013, and 2015. Exercise 5(D) carries 2005, 2006, 2007, 2010, 2012, and 2014. These tags signal question types the board repeats, so practise those first.
Browse the other chapters in the Class 10 Maths book list or view the full Class 10 books collection. Check the official CISCE website for the Class 10 Mathematics syllabus and examination details.
Frequently asked questions
How can I check whether the roots of a quadratic equation are real without solving it?
Use the discriminant D = b^2 - 4ac. If D \geq 0, the roots are real. If D \lt 0, the roots are imaginary.
What does it mean if a quadratic equation has equal roots, and how do I find the missing coefficient?
Equal roots mean D = 0. Set b^2 - 4ac = 0 and solve for the missing coefficient. For x^2 - 10x + k = 0, you need 100 - 4k = 0, so k = 25.
When should I use factorisation instead of the quadratic formula?
Use factorisation when the left side factors cleanly. The formula works for every quadratic, including ones that do not factorise easily.
How do I solve an equation such as x^4 - 5x^2 + 4 = 0?
Substitute y = x^2. The equation becomes y^2 - 5y + 4 = 0, which factorises to (y - 1)(y - 4) = 0. Then y = 1 or 4, giving x = \pm 1 or \pm 2.
Why does the quadratic formula contain a \pm sign?
Taking a square root yields two values, one positive and one negative. The \pm sign captures both roots of the equation.
Reference: ICSE Class 10 Mathematics textbook material, chapter 5: Quadratic Equations — CISCE-aligned.
More from this book
Explore Class 10 Maths Notes
Related chapters: