Is it correct to say: Forgetting the ± in the quadratic formula (writing only one root)?

Both signs give valid roots. Write BOTH values, then apply the physical restriction if any.

Is it correct to say: Cancelling x from both sides of x² = 5x?

Cancelling loses the root x = 0. Instead: x² − 5x = 0 ⇒ x(x−5) = 0 ⇒ x = 0 or 5.

Is it correct to say: Assuming D = 0 gives two different roots?

D = 0 gives ONE repeated root x = −b/(2a). Count multiplicity 2 only if asked.

Is it correct to say: Accepting negative or non-integer roots in real-world problems (speed, ages)?

Reject roots that violate the physical setup. Speed > 0; age is a positive integer; number of items is a whole number.

Is it correct to say: Writing the wrong sign for a in sum/product of roots?

Formula uses the STANDARD form ax² + bx + c = 0 with 'a' the coefficient of x². Move everything to one side first.

Is it correct to say: Confusing 'equal roots' with 'no roots' when D = 0?

D = 0 ⇒ EQUAL real roots (two coincident). D < 0 ⇒ NO real roots.

Quadratic Equations

Standard form, factorisation, quadratic formula, discriminant, nature of roots and word problems — NCERT Class 10 Maths Ch 4

📐 9 formulas✏️ 3 examples🎯 6 practice⚖️ 5-6 marks🏫 CBSE📚 Class 10✓ 2025–26 syllabus

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Board Exam Tips

→Quadratic formula derivation (by completing the square) — a 3-mark HOTS question repeatedly asked.
→Nature of roots: use D = b² − 4ac. D > 0 real distinct, D = 0 real equal, D < 0 no real.
→Word problems on speed/time/age/area — reduce carefully to a quadratic before solving.
→Verify your two roots by substituting BOTH back into the equation before writing the answer.
→Discriminant condition k for equal roots is a 2-mark question every year.

📐 Formulas(9)

Standard Form

a x^{2} + b x + c = 0, a \neq = 0

Quadratic Formula★ Board fav

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Discriminant★ Board fav

D = b^{2} - 4 a c

Nature: Two Real Distinct Roots

D > 0

Nature: Two Equal Real Roots

D = 0

Nature: No Real Roots

D < 0

Sum and Product of Roots

α + β = - \frac{b}{a}, α β = \frac{c}{a}

Reconstructing a Quadratic from Roots

x^{2} - (α + β) x + α β = 0

Completing the Square Identity

x^{2} + 2 p x + q = (x + p)^{2} + (q - p^{2})

✏️ Solved Examples

1Solved Exampleeasy3 steps

Solve x² − 5x + 6 = 0 by factorisation.

Find two numbers whose sum is −5 and product is 6

- 2 + - 3 = - 5, (- 2) (- 3) = 6

2Solved Exampleboard3 steps

Find the value(s) of k for which the equation 2x² + kx + 8 = 0 has equal roots.

For equal roots, D = 0

b^{2} - 4 a c = 0

3Solved ExampleHOTS6 steps

A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, the journey would have taken 1 hour less. Find the original speed of the train.

Let original speed = x km/h. Time = distance/speed

t = \frac{360}{x}

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Forgetting the ± in the quadratic formula (writing only one root)
✓Both signs give valid roots. Write BOTH values, then apply the physical restriction if any.
2
Cancelling x from both sides of x² = 5x
✓Cancelling loses the root x = 0. Instead: x² − 5x = 0 ⇒ x(x−5) = 0 ⇒ x = 0 or 5.
3
Assuming D = 0 gives two different roots
✓D = 0 gives ONE repeated root x = −b/(2a). Count multiplicity 2 only if asked.
4
Accepting negative or non-integer roots in real-world problems (speed, ages)
✓Reject roots that violate the physical setup. Speed > 0; age is a positive integer; number of items is a whole number.
5
Writing the wrong sign for a in sum/product of roots
✓Formula uses the STANDARD form ax² + bx + c = 0 with 'a' the coefficient of x². Move everything to one side first.
6
Confusing 'equal roots' with 'no roots' when D = 0
✓D = 0 ⇒ EQUAL real roots (two coincident). D < 0 ⇒ NO real roots.

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
Solve by factorisation: x² + 7x + 10 = 0.
Q2
Solve using the quadratic formula: 2x² − 7x + 3 = 0.
Q3
Find the nature of roots of x² + 5x + 7 = 0.
Q4
The product of two consecutive positive integers is 306. Find them.
Q5
For what value of k does 3x² − 2kx + 12 = 0 have equal roots?
Q6
The sum of two numbers is 15 and the sum of their reciprocals is 3/10. Find the numbers.

📝 Notes

Quadratic Equations

Any equation of the form ax² + bx + c = 0 (a ≠ 0) is a quadratic equation. It has at most two real roots — the x-values that satisfy the equation.

Three methods to solve

Factorisation — split the middle term.
Completing the square — rewrite as (x + p)² = k.
Quadratic formula — plug into x = [−b ± √(b² − 4ac)]/(2a).

Formula works ALWAYS; factorisation works only when integer/rational roots exist.

Discriminant D = b² − 4ac decides everything

| D | Nature of roots | Graph of y = ax² + bx + c | |---|---|---| | D > 0 | Two distinct real roots | Crosses x-axis at 2 points | | D = 0 | Two equal (coincident) real roots | Touches x-axis at 1 point | | D < 0 | No real roots | Does not touch x-axis |