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CBSE Class 10 · Maths · board exam 2027

Chapter 4: Quadratic Equations — notes

The chapter’s 6 parts in the book’s order — the first idea of each and the points Medha Ma’am writes on the board — and the summary at the end. The full lesson and a check question for each part are in the app.

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  1. 1. What is a quadratic equation

    A charity trust wants to build a prayer hall.

    • • Prayer hall: area = 300 m², length = 2 × breadth + 1
    • • Let breadth = x m
    • • Length = (2x + 1) m
    • • Area = (2x + 1) × x
    • • = 2x² + x
    • • 2x² + x = 300
    • • 2x² + x − 300 = 0
    • • Quadratic equation: ax² + bx + c = 0, a ≠ 0
    • • a, b, c are real numbers
    • • Standard form: x² term, then x term, then constant
    • • If a = 0: 0 × x² + bx + c = 0 ⇒ bx + c = 0 (linear)
    • • 2x² + x − 300 = 0: a = 2, b = 1, c = −300

  2. 2. Checking whether an equation is quadratic

    Sometimes an equation is not written in the standard form.

    • • Rule: open brackets → bring all terms to one side → simplify → is x² the highest power?
    • • Example 1: (x − 2)² + 1 = 2x − 3
    • • (x − 2)² = x² − 4x + 4
    • • LHS = x² − 4x + 4 + 1 = x² − 4x + 5
    • • x² − 4x + 5 − 2x + 3 = 0
    • • x² − 6x + 8 = 0
    • • a = 1 ≠ 0 ⇒ quadratic ✓
    • • Example 2: x(x + 1) + 8 = (x + 2)(x − 2)
    • • LHS = x² + x + 8
    • • RHS = x² − 4
    • • x² + x + 8 = x² − 4
    • • x + 8 + 4 = 0
    • • x + 12 = 0 ⇒ not quadratic ✗
    • • Example 3: (x + 2)³ = x³ − 4
    • • (a + b)³ = a³ + 3a²b + 3ab² + b³
    • • (x + 2)³ = x³ + 6x² + 12x + 8
    • • x³ + 6x² + 12x + 8 = x³ − 4
    • • 6x² + 12x + 8 + 4 = 0
    • • 6x² + 12x + 12 = 0 ⇒ quadratic ✓

  3. 3. Roots and solving by factorisation

    We find its roots.

    • • α is a root of ax² + bx + c = 0 if aα² + bα + c = 0
    • • A quadratic equation has at most two roots
    • • Example: 2x² − 5x + 3 = 0
    • • a × c = 2 × 3 = 6
    • • Product 6, sum −5: (−2) × (−3) = 6, (−2) + (−3) = −5
    • • 2x² − 2x − 3x + 3 = 0
    • • 2x(x − 1) − 3(x − 1) = 0
    • • (2x − 3)(x − 1) = 0
    • • 2x − 3 = 0 ⇒ 2x = 3 ⇒ x = 3/2
    • • x − 1 = 0 ⇒ x = 1
    • • Roots: 3/2 and 1
    • • Check x = 1: 2(1)² − 5(1) + 3 = 2 − 5 + 3 = 0 ✓
    • • Prayer hall: 2x² + x − 300 = 0
    • • a × c = 2 × (−300) = −600
    • • 25 × (−24) = −600, 25 + (−24) = 1
    • • 2x² + 25x − 24x − 300 = 0
    • • x(2x + 25) − 12(2x + 25) = 0
    • • (x − 12)(2x + 25) = 0
    • • x = 12 or x = −25/2 = −12.5
    • • Breadth cannot be negative ⇒ reject x = −12.5
    • • Breadth = 12 m, Length = 2 × 12 + 1 = 25 m
    • • Check: 25 × 12 = 300 m² ✓
    • • Sum of roots = −b/a, product of roots = c/a
    • • 2x² − 5x + 3 = 0: sum = 5/2, product = 3/2
    • • 1 + 3/2 = 5/2 ✓

  4. 4. The quadratic formula

    Splitting the middle term is quick, but sometimes we cannot find the two numbers at all.

    • • Goal: a formula for every ax² + bx + c = 0
    • • ax² + bx + c = 0
    • • x² + b/ax + c/a = 0 (÷ a, a ≠ 0)
    • • x² + b/ax = −c/a
    • • Pattern: (x + k)² = x² + 2kx + k²
    • • 2k = b/a ⇒ k = b/2a
    • • x² + b/ax + (b/2a)² = (b/2a)² − c/a
    • • (x + b/2a)² = b²/4a² − c/a
    • • c/a = 4ac/4a²
    • • (x + b/2a)² = (b² − 4ac)/4a²
    • • If b² − 4ac ≥ 0:
    • • x + b/2a = ±(√(b² − 4ac))/2a
    • • x = (−b ± √(b² − 4ac))/2a
    • • Example: 2x² + x − 528 = 0
    • • a = 2, b = 1, c = −528
    • • b² = 1² = 1
    • • 4ac = 4 × 2 × (−528) = −4224
    • • b² − 4ac = 1 − (−4224) = 1 + 4224 = 4225
    • • √4225 = 65 (65 × 65 = 4225)
    • • −b = −1, 2a = 2 × 2 = 4
    • • x = (−1 ± 65)/4
    • • Taking +: x = (−1 + 65)/4 = 64/4 = 16
    • • Taking −: x = (−1 − 65)/4 = (−66)/4 = −33/2
    • • Breadth cannot be negative ⇒ reject x = −33/2
    • • Breadth = 16 m, Length = 2 × 16 + 1 = 33 m
    • • Check: 33 × 16 = 528 ✓
    • • Easy split → factorise; hard split → formula

  5. 5. Nature of roots: the discriminant

    Look at the formula again.

    • • Discriminant: D = b² − 4ac
    • • x = (−b ± √D)/2a
    • • D > 0 ⇒ two distinct real roots
    • • D = 0 ⇒ two equal real roots, x = −b/2a
    • • D < 0 ⇒ no real roots
    • • D > 0 and a perfect square ⇒ rational roots
    • • D > 0, not a perfect square ⇒ irrational roots
    • • Example: 2x² − 4x + 3 = 0
    • • a = 2, b = −4, c = 3
    • • b² = (−4)² = 16
    • • 4ac = 4 × 2 × 3 = 24
    • • D = b² − 4ac = 16 − 24 = −8
    • • D < 0 ⇒ no real roots
    • • Example: 2x² + kx + 3 = 0 has equal roots. Find k.
    • • a = 2, b = k, c = 3
    • • Equal roots ⇒ D = 0
    • • D = k² − 4 × 2 × 3 = k² − 24
    • • k² − 24 = 0 ⇒ k² = 24
    • • k = ±√24 = ±2√6 (√24 = √(4 × 6) = 2√6)

  6. 6. Word problems, step by step

    Most five-mark questions in this chapter are word problems.

    • • Let → equation from the words → standard form → solve → reject with reason, check, units
    • • A train travels 360 km at a uniform speed. If the speed were 5 km/h more, it would take 1 hour less. Find the speed.
    • • Let speed = x km/h, faster speed = (x + 5) km/h
    • • Time = distance/speed
    • • Usual time = 360/x h, faster time = 360/(x + 5) h
    • • 360/x − 360/(x + 5) = 1
    • • (360(x + 5) − 360x)/(x(x + 5)) = 1
    • • 1800/(x(x + 5)) = 1
    • • 1800 = x² + 5x
    • • x² + 5x − 1800 = 0
    • • 45 × (−40) = −1800, 45 + (−40) = 5
    • • x² + 45x − 40x − 1800 = 0
    • • x(x + 45) − 40(x + 45) = 0
    • • (x + 45)(x − 40) = 0 ⇒ x = −45 or x = 40
    • • Speed cannot be negative ⇒ reject x = −45
    • • Check: 360/40 − 360/45 = 9 − 8 = 1 ✓
    • • Speed of the train = 40 km/h

Summary

Very good! You worked hard today. A quadratic equation is a x² plus b x plus c equals zero, with a not zero. And before deciding, we always simplify fully. Then we solved by splitting the middle term, using the idea that if a product is zero, one factor is zero. We built the quadratic formula by completing the square. And the discriminant, b² minus 4ac, told us the nature of the roots. Write the formula and the three cases of the discriminant in your copy once, without looking. Your homework is on your screen. In word problems, never forget to reject a negative speed, age or length, with its reason. If anything is not clear, press the Ask button and ask me. Done today!

AskMedha · Medha Ma’am — CBSE Class 10 Maths, Chapter 4. Written from the textbook, checked twice.

Questions parents and students ask

How do I get the "Quadratic Equations" notes as a PDF?
Press "Save as PDF" — the phone’s print sheet opens; choose "Save as PDF" there. The PDF holds the notes only.
Where are these notes written from?
From the chapter’s own textbook text, part by part, in the book’s own terms; a second, independent model then checked every part against the book. The full lesson, a check question for each part and the practice are in the AskMedha app.

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