1
Which of these is a quadratic equation?
💡 Hint:
Degree exactly 2 and a ≠ 0.
Explanation:
2x² − 5x + 3 = 0 is quadratic.
Class 10 · Maths · Chapter 4 · बिहार बोर्ड (BSEB)CBSE · NCERT 2026-27
Quadratic Equations
Quadratic Equations
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- 1 Standard form ax² + bx + c = 0
- 2 Roots by factorisation
- 3 Completing the square
- 4 The quadratic formula
- 5 The discriminant and the nature of the roots
- 6 A word problem and the root that fits
- Chapter winner — every lesson at mastery ★
🎬 Shorts
This chapter has no Short on the channel yet, so ten empty cards are not shown. The notes, diagrams and checks are complete.
📄 PDF download — notes + answer key
Standard form ax² + bx + c = 0
मानक रूप ax² + bx + c = 0 · NCERT 4.2 · Exercise 4.1
a cannot be zeroNotes
A quadratic equation has degree 2. The standard form is ax² + bx + c = 0. a, b, c are real and a ≠ 0. b or c may be zero. a may not.
Setting a polynomial equal to zero makes an equation. The zeroes of ax² + bx + c and the roots of ax² + bx + c = 0 are the same. If the x² term drops, it is no longer quadratic.
Bring every term to the leftActivity
Exercise 4.1 asks you to decide whether an equation is quadratic. Expand (x + 2)² = 9: x² + 4x + 4 = 9, then x² + 4x − 5 = 0. Now a = 1, b = 4, c = −5. If no x² remains after expanding, do not call it quadratic.
Worked exampleExample
Question: Write 2x² = 5x − 3 in standard form and name a, b, c.
Formula: ax² + bx + c = 0.
Substitution: 2x² − 5x + 3 = 0.
Answer: a = 2, b = −5, c = 3. These are exact values. a ≠ 0, so it is quadratic.
10-second revision
- Degree 2 and a ≠ 0
- All terms on the left, right side 0
- Roots and the zeroes of the polynomial are the same
Board tip · BSEBBoard tip
In a BSEB answer write a, b and c with signs. Moving 5x to the left makes −5x.
Board tip · CBSEBoard tip
On a CBSE recognition item, give a one-line reason: a ≠ 0, or the x² term is missing.
Check your understandingall correct = mastery ★
2
In ax² + bx + c = 0, a may be zero.
Explanation:
False — a must satisfy a ≠ 0.
3
In standard form the right side is ______.
💡 Hint:
Every term is on the left.
Explanation:
0.
4
The standard form of (x + 2)² = 9 is —
💡 Hint:
Expand, then subtract 9.
Explanation:
x² + 4x + 4 − 9 = 0, that is x² + 4x − 5 = 0.
5
Write x² + 3 = 4x in standard form and name a, b, c.
Model answer (board format)
Formula: ax² + bx + c = 0. Substitution: x² − 4x + 3 = 0. Exact values a = 1, b = −4, c = 3.
Check your answer:
Roots by factorisation
गुणनखंड से मूल · NCERT 4.3 · Exercise 4.2
If a product is zero, one factor is zeroNotes
For x² + bx + c = 0, find two numbers whose product is c and whose sum is b. Then (x − p)(x − q) = 0. Set each factor equal to zero on its own.
If a is not 1, split the middle term. In 2x² + 7x + 3, 7 = 6 + 1, and 6 × 1 = 2 × 3. Exercise 4.2 of the current NCERT is this method.
Split the middle term and make bracketsActivity
Write 2x² + 7x + 3 = 0 as 2x² + 6x + x + 3 = 0. Then 2x(x + 3) + 1(x + 3) = 0, so (2x + 1)(x + 3) = 0. Now 2x + 1 = 0 or x + 3 = 0. Put both roots back into the equation and check.
Worked exampleExample
Question: Find the roots of x² − 5x + 6 = 0 by factorisation.
Formula: Two numbers with product +6 and sum −5.
Substitution: −2 and −3. (x − 2)(x − 3) = 0.
Answer: x = 2 or x = 3. Exact roots. Check: 4 − 10 + 6 = 0 and 9 − 15 + 6 = 0.
10-second revision
- Product c and sum b
- Set each factor equal to zero
- Check both roots in the original equation
Board tip · BSEBBoard tip
In a BSEB answer the split of the middle term should be visible. Do not jump to the brackets.
Board tip · CBSEBoard tip
On CBSE write both roots. One root is an incomplete answer.
Check your understandingall correct = mastery ★
1
The roots of x² − 5x + 6 = 0 are —
💡 Hint:
(x − 2)(x − 3).
Explanation:
x = 2 and x = 3.
2
(x − 2)(x − 3) = 0 means both factors must be zero together.
Explanation:
False — it is enough that one factor is zero.
3
One root of 2x² + 7x + 3 = 0 is x = ______.
💡 Hint:
(2x + 1)(x + 3) = 0.
Explanation:
x = −3 or x = −1/2.
4
Solve x² − 7x + 12 = 0 by factorisation.
Model answer (board format)
The numbers −3 and −4 have product 12 and sum −7. (x − 3)(x − 4) = 0. Exact roots x = 3 and x = 4.
Check your answer:
Completing the square
पूर्ण वर्ग बनाना · Older Bihar exercise 4.3 · the root of the formula
The square of half the coefficient, on both sidesNotes
Add (b/2)² to the x² + bx terms, on both sides. The left side becomes (x + b/2)². Then take square roots and write both signs of ±.
This method is exercise 4.3 of the older Bihar book. The current NCERT has no separate exercise for it, but the quadratic formula comes by this same path. If a is not 1, divide the whole equation by a first.
Halve the coefficient and addActivity
In x² + 6x + 5 = 0 the coefficient of x is 6. Half of it is 3, and 3² = 9. Move the constant to the right, then add 9. The left side becomes (x + 3)². Adding only on the left spoils the equation.
Worked exampleExample
Question: Solve x² + 6x + 5 = 0 by completing the square.
Formula: (x + b/2)² = (b/2)² − c, here b = 6.
Substitution: x² + 6x = −5. (x + 3)² = 9 − 5 = 4. x + 3 = ±2.
Answer: x = −3 + 2 = −1 or x = −3 − 2 = −5. Exact roots. Check: 1 − 6 + 5 = 0 and 25 − 30 + 5 = 0.
10-second revision
- Add the square of half the coefficient on both sides
- The ± gives both roots
- If a ≠ 1, divide by a first
Board tip · BSEBBoard tip
In exercise 4.3 of the older BSEB book, the line (x + ...)² should stand on its own.
Board tip · CBSEBoard tip
Current CBSE exercise 4.3 asks for the nature of the roots. Completing the square is not asked there as a separate method.
Check your understandingall correct = mastery ★
1
To complete the square on x² + 6x, you add —
💡 Hint:
Half is 3.
Explanation:
3² = 9. (x + 3)² = x² + 6x + 9.
2
In completing the square the added number joins only the left side.
Explanation:
False — it is added on both sides.
3
One root of x² + 6x + 5 = 0 is x = ______.
💡 Hint:
x + 3 = ±2.
Explanation:
x = −1 or x = −5.
4
From (x + 3)² = 4 the values of x are —
💡 Hint:
Write ±2.
Explanation:
x + 3 = 2 or x + 3 = −2, so −1 and −5.
5
If a = 2, dividing the equation by 2 before completing the square is correct.
Explanation:
True — first make the coefficient of x² equal to 1.
6
Solve x² + 4x − 5 = 0 by completing the square.
Model answer (board format)
x² + 4x = 5. (x + 2)² = 4 + 5 = 9. x + 2 = ±3. x = 1 or x = −5. Check: 1 + 4 − 5 = 0.
Check your answer:
The quadratic formula
द्विघात सूत्र · x = [−b ± √(b² − 4ac)] / (2a)
The formula linked with SridharacharyaNotes
Running completing the square on ax² + bx + c = 0 produces the formula. x = [−b ± √(b² − 4ac)] / (2a). The denominator is 2a, not just 2.
b is the coefficient in the equation, sign included. Find D = b² − 4ac on its own first. If D is negative, do not write real roots. The current book keeps this formula in the section on the nature of the roots.
Copy a, b, c into a small tableActivity
For 2x² − 7x + 3 = 0 write a = 2, b = −7, c = 3. Then −b = 7. D = (−7)² − 4·2·3. After the square root, make two lines from the ± and divide each by 2a = 4.
Worked exampleExample
Question: Solve 2x² − 7x + 3 = 0 by the formula.
Formula: x = [−b ± √(b² − 4ac)] / (2a).
Substitution: a = 2, b = −7, c = 3. D = 49 − 24 = 25. x = [7 ± 5] / 4.
Answer: x = 12/4 = 3 or x = 2/4 = 1/2. Exact roots. Check: 2·9 − 7·3 + 3 = 18 − 21 + 3 = 0.
10-second revision
- The denominator is 2a
- −b includes the sign of b
- Find D first, then the ±
Board tip · BSEBBoard tip
In a BSEB answer the line of a, b, c and D should come before the formula.
Board tip · CBSEBoard tip
On CBSE you may write 1/2 as 0.5, but the fraction 1/2 stays clearer.
Check your understandingall correct = mastery ★
1
In 2x² − 7x + 3 = 0, −b equals —
💡 Hint:
b = −7.
Explanation:
−b = −(−7) = 7.
2
The denominator of the quadratic formula is always 2.
Explanation:
False — the denominator is 2a.
3
The larger root of 2x² − 7x + 3 = 0 is x = ______.
💡 Hint:
(7 + 5)/4.
Explanation:
x = 3. The other root is 1/2.
4
Inside D = b² − 4ac the sign in front of 4ac is —
💡 Hint:
The formula is b² − 4ac.
Explanation:
4ac is subtracted. If c itself is negative, subtracting it looks like adding.
5
Solve x² − 4x + 3 = 0 by the formula.
Model answer (board format)
a = 1, b = −4, c = 3. D = 16 − 12 = 4. x = [4 ± 2] / 2. x = 3 or x = 1.
Check your answer:
The discriminant and the nature of the roots
विविक्तकर और मूलों की प्रकृति · NCERT 4.4 · Exercise 4.3 (current book)
D tells how many roots there areNotes
D = b² − 4ac. Keep the three cases.
| D | Roots |
|---|---|
| D > 0 | Two different real |
| D = 0 | Two equal real, −b/(2a) |
| D < 0 | No real root |
When D = 0 there are still two roots, and they are equal. When D < 0 do not write a real number. This is exercise 4.3 of the current NCERT. In the older book it is 4.4.
First D, then the name of the caseActivity
If the question asks for the nature, look at the sign of D before finding roots. If D = 0, compute x = −b/(2a) once and write that both roots are equal. If D < 0, do not try the square root.
Worked exampleExample
Question: State the nature of x² − 6x + 9 = 0 and write the roots.
Formula: D = b² − 4ac. If D = 0, then x = −b/(2a).
Substitution: a = 1, b = −6, c = 9. D = 36 − 36 = 0. x = 6/2 = 3.
Answer: Two equal real roots, both x = 3. Check: 9 − 18 + 9 = 0.
10-second revision
- D > 0: two different real roots
- D = 0: two equal roots
- D < 0: no real root
Board tip · BSEBBoard tip
In a BSEB answer write the full arithmetic of D, not only the name of the case.
Board tip · CBSEBoard tip
On CBSE do not call D = 0 “one root”. The book says two equal real roots.
Check your understandingall correct = mastery ★
1
D for x² + x + 1 = 0 is —
💡 Hint:
1 − 4.
Explanation:
D = 1 − 4 = −3. No real root.
2
D = 0 means there is no root.
Explanation:
False — there are two equal real roots.
3
If D < 0, the number of real roots is ______.
💡 Hint:
A negative number has no real square root.
Explanation:
Zero.
4
The roots of x² − 4x + 3 = 0 on the number line are —
💡 Hint:
D = 4 > 0.
Explanation:
x = [4 ± 2] / 2, so 3 and 1.
5
If D > 0 the roots are real and different.
Explanation:
True — two different real roots.
6
State the nature of x² + 2x + 5 = 0 from D. Do not invent roots.
Model answer (board format)
a = 1, b = 2, c = 5. D = 4 − 20 = −16 < 0. There is no real root.
Check your answer:
A word problem and the root that fits
शब्द-समस्या और शर्त वाला मूल · Area · a positive length
Find both roots, then read the conditionNotes
A length, a side or a count of objects is not negative. The formula will give both roots. Drop the root that breaks the condition, but show that it also appeared.
Area is in square metres. Keep the same unit on both sides when you build the equation. The area of a rectangle is length times width.
The equation first, then the unitActivity
Let the width be w metres and the length 4 metres more than the width. The area is 45 square metres. Write w(w + 4) = 45. Make it w² + 4w − 45 = 0. Only the positive root is a side.
Worked exampleExample
Question: A rectangle has width w metres and length w + 4 metres. The area is 45 m². Find the sides.
Formula: w(w + 4) = 45, then x = [−b ± √(b² − 4ac)] / (2a).
Substitution: w² + 4w − 45 = 0. a = 1, b = 4, c = −45. D = 16 + 180 = 196 = 14². w = [−4 ± 14] / 2.
Answer: w = 5 or w = −9. A length is not negative, so the width is 5 m and the length is 9 m. Check: 5 × 9 = 45 m².
10-second revision
- Read the condition: length is positive
- Show the rejected root as well
- Check with the unit included
Board tip · BSEBBoard tip
In a BSEB answer write why a root is rejected, do not only leave the positive number.
Board tip · CBSEBoard tip
On CBSE write the unit m² with the area. Only 45 stays incomplete.
Check your understandingall correct = mastery ★
1
The root of w² + 4w − 45 = 0 that can be a side is —
💡 Hint:
A length is not negative.
Explanation:
w = 5 m. −9 m is rejected.
2
In a word problem both mathematical roots are always acceptable.
Explanation:
False — a root that breaks the condition is dropped.
3
If the width is 5 m and the length is 9 m, the area is ______ m².
💡 Hint:
5 × 9.
Explanation:
45 m².
4
The product of two consecutive positive integers is 72. Find the integers.
Model answer (board format)
n(n + 1) = 72. n² + n − 72 = 0. D = 1 + 288 = 289 = 17². n = (−1 ± 17)/2. n = 8 or n = −9. The positive pair is 8 and 9. Check: 8 × 9 = 72.
Check your answer:
❓ Full question bank — with answers and explanations — 64 questions
Pick a type. The 30 lesson checks are separate — each lesson has as many as its topic needs. All correct earns mastery ★.
BSEB modelpractice · not annual60 tried · 0 correctMultiple choiceMCQ120 tried · 0 correctTrue / falseTrue / false80 tried · 0 correctFill in the blanksFill80 tried · 0 correctMatchMatch20 tried · 0 correctAssertion–reasonAssertion–reason50 tried · 0 correctBalancingBalancing40 tried · 0 correctIdentifyIdentify20 tried · 0 correctShort and longVSA, SA, LA110 tried · 0 correctCBSE-stylenot a PYQ60 tried · 0 correct
No question is marked as a verified past paper. The BSEB set is a model for practice. CBSE items are CBSE-style, not a copy of any year’s paper.
Multiple choice
0/12Pick one option. A wrong try brings a hint.
1
In ax² + bx + c = 0 it is required that —
💡 Hint:
If a is zero the degree drops.
Explanation:
a ≠ 0. b and c may be zero.
2
The roots of x² − 7x + 12 = 0 are —
💡 Hint:
3 × 4 = 12 and 3 + 4 = 7.
Explanation:
x = 3 and x = 4.
3
The number added to complete the square on x² + 6x is —
💡 Hint:
(6/2)².
Explanation:
9.
4
D for 2x² − 7x + 3 = 0 is —
💡 Hint:
49 − 24.
Explanation:
D = 25.
5
D < 0 means —
💡 Hint:
The square root of a negative is not real.
Explanation:
No real root.
6
The roots of x² − 6x + 9 = 0 are —
💡 Hint:
D = 0.
Explanation:
Two equal roots, both 3.
7
The denominator of the formula is —
💡 Hint:
a stays with it.
Explanation:
2a.
8
x² + x + 1 = 0 has —
💡 Hint:
D = −3.
Explanation:
D < 0, no real root.
9
w = −9 m is rejected as a side because —
💡 Hint:
Read the condition.
Explanation:
A length must be positive.
10
One root from (x − 2)(x − 3) = 0 is —
💡 Hint:
A factor is zero.
Explanation:
x = 2 or x = 3.
11
The roots of x² + 4x − 5 = 0 are —
💡 Hint:
(x + 5)(x − 1).
Explanation:
x = 1 and x = −5.
12
The formula for two equal roots is —
💡 Hint:
D = 0.
Explanation:
Both roots are −b/(2a).
True or false
0/81
The zeroes of the polynomial ax² + bx + c are the roots of the equation.
Explanation:
True — they are the same values.
2
Writing only one root in factorisation is a complete answer.
Explanation:
False — write both roots, unless they are equal and you say so.
3
When D > 0 the roots are real and different.
Explanation:
True.
4
In completing the square it is fine to drop the negative sign of ±.
Explanation:
False — both signs can give two roots.
5
If c is negative, −4ac adds a positive quantity.
Explanation:
True — a minus times a minus is a plus.
6
x + 5 = 0 is a quadratic equation.
Explanation:
False — the degree is 1.
7
When D = 0 the roots are real.
Explanation:
True — two equal real roots.
8
Exercise 4.2 of the current NCERT is completing the square.
Explanation:
False — 4.2 is factorisation. Completing the square is exercise 4.3 of the older Bihar book.
Fill in the blanks
0/81
The discriminant is D = b² − ______.
💡 Hint:
4ac.
Explanation:
Four a c.
2
If a is not nonzero, the equation does not stay ______.
💡 Hint:
Quadratic.
Explanation:
Degree 2.
3
If D = 0 both roots are ______.
💡 Hint:
Equal.
Explanation:
The same value.
4
The smaller root of x² − 5x + 6 = 0 is ______.
💡 Hint:
2.
Explanation:
The smaller of 2 and 3.
5
The sign in front of the square root in the formula is ______.
💡 Hint:
±.
Explanation:
Both roots.
6
5 m × 9 m = ______ m².
💡 Hint:
45.
Explanation:
The area.
7
x² + 4x + ______ = (x + 2)².
💡 Hint:
4.
Explanation:
Half the coefficient is 2.
8
If there is no real root, D is ______ 0.
💡 Hint:
D < 0.
Explanation:
Less than zero.
Match
0/21
Match the case of D with the roots.
Column B: A. Two equal roots · B. Not quadratic · C. No real root · D. Two different real roots
1. D > 0
2. D = 0
3. D < 0
4. a = 0
Explanation:
D > 0 means different roots, D = 0 equal roots, D < 0 no real root, and a = 0 is not quadratic.
2
Match the method with its mark.
Column B: A. The sign of D · B. (x − p)(x − q) = 0 · C. (x + b/2)² · D. [−b ± √D] / (2a)
1. Factorisation
2. Completing the square
3. Formula
4. Nature
Explanation:
Factorisation is the brackets, completing the square is the square, the formula has ±, and the nature is the sign of D.
Assertion–reason
0/5Check both statements, then see whether the reason explains the assertion.
1
Assertion (A): x² − 6x + 9 = 0 has two equal real roots.
Reason (R): Its D = 36 − 36 = 0.
Explanation:
Both are true and R explains A.
2
Assertion (A): When D > 0 there are two different real roots.
Reason (R): In ax² + bx + c = 0, a is not zero.
Explanation:
Both are true, but R does not explain this case.
3
Assertion (A): x² + x + 1 = 0 has no real root.
Reason (R): Its D is 5.
Explanation:
A is true. R is false, D = 1 − 4 = −3.
4
Assertion (A): x + 4 = 0 is a quadratic equation.
Reason (R): In the standard form of a quadratic, the coefficient of x² is not zero.
Explanation:
A is false, the degree is 1. R is true.
5
Assertion (A): The roots of 2x² − 7x + 3 = 0 are 3 and 1/2.
Reason (R): D = 25 and x = [7 ± 5] / 4.
Explanation:
Both are true and R explains the calculation.
Coefficient practice
0/4These chemistry lines are not a result of this maths chapter. They are only practice in filling coefficients. A blank means 1.
1
This is coefficient practice and not a result of this maths chapter. In the line where H2 + O2 makes water, Fill coefficients (blank = 1).
H2 + O2 → H2O
Explanation: 2H2 + O2 → 2H2O
2
This is coefficient practice and not a result of this maths chapter. In the line where N2 + H2 makes ammonia, Fill coefficients (blank = 1).
N2 + H2 → NH3
Explanation: N2 + 3H2 → 2NH3
3
This is coefficient practice and not a result of this maths chapter. In the line where C + O2 makes carbon dioxide, Fill coefficients (blank = 1).
C + O2 → CO2
Explanation: C + O2 → CO2
4
This is coefficient practice and not a result of this maths chapter. In the burning line of CH4 + O2, Fill coefficients (blank = 1).
CH4 + O2 → CO2 + H2O
Explanation: CH4 + 2O2 → CO2 + 2H2O
Classify
0/21
Place each equation in the case of D.
x² − 4x + 3 = 0
x² − 6x + 9 = 0
x² + x + 1 = 0
2x² − 7x + 3 = 0
Explanation:
The first has D = 4, the second 0, the third −3 and the fourth 25.
2
Place each equation as quadratic or not.
x² + 1 = 0
3x + 1 = 0
x³ − x = 0
5x² = 20
Explanation:
x² + 1 = 0 and 5x² − 20 = 0 are quadratic. Degree 1 and degree 3 are not.
Very short answer
0/41
Write the standard form of a quadratic equation.
Model answer (board format)
ax² + bx + c = 0, where a ≠ 0.
Check your answer:
2
Write the definition of the discriminant.
Model answer (board format)
D = b² − 4ac. This decides the nature of the roots.
Check your answer:
3
Write the three cases of D, one line each.
Model answer (board format)
D > 0 means two different real roots. D = 0 means two equal real roots. D < 0 means no real root.
Check your answer:
4
Write the quadratic formula.
Model answer (board format)
x = [−b ± √(b² − 4ac)] / (2a), a ≠ 0.
Check your answer:
Short answer
0/41
Solve x² − 8x + 15 = 0 by factorisation.
Model answer (board format)
(x − 3)(x − 5) = 0 because 3 + 5 = 8 and 3 × 5 = 15. The roots are x = 3 and x = 5.
Check your answer:
2
Solve x² + 8x + 7 = 0 by completing the square.
Model answer (board format)
x² + 8x = −7. (x + 4)² = 16 − 7 = 9. x + 4 = ±3. x = −1 or x = −7.
Check your answer:
3
Find the nature and the roots of 3x² − 5x + 2 = 0 by the formula.
Model answer (board format)
a = 3, b = −5, c = 2. D = 25 − 24 = 1 > 0, two different real roots. x = [5 ± 1] / 6. x = 1 or x = 4/6 = 2/3.
Check your answer:
4
A square has side x metres. x² = 49. Write the side and the area with units.
Model answer (board format)
x² − 49 = 0, (x − 7)(x + 7) = 0. x = 7 or x = −7. The side is 7 m. The area is 49 m².
Check your answer:
Long answer
0/31
Solve x² − 5x + 6 = 0 both by factorisation and by the formula. Match the two answers.
Model answer (board format)
Factorisation: (x − 2)(x − 3) = 0, x = 2 or 3. Formula: a = 1, b = −5, c = 6, D = 25 − 24 = 1, x = [5 ± 1] / 2. x = 3 or x = 2. Both give the same exact roots.
Check your answer:
2
State the nature of these three from D: (i) x² − 4x + 3 = 0 (ii) x² − 6x + 9 = 0 (iii) x² + x + 1 = 0. Where the roots are real, write them too.
Model answer (board format)
(i) D = 16 − 12 = 4 > 0, roots 3 and 1. (ii) D = 36 − 36 = 0, both roots 3. (iii) D = 1 − 4 = −3 < 0, no real root.
Check your answer:
3
A garden is 3 m less in width than in length. The area is 40 m². Find the sides. Why is a negative root dropped?
Model answer (board format)
Let the length be x m and the width x − 3. x(x − 3) = 40. x² − 3x − 40 = 0. D = 9 + 160 = 169 = 13². x = [3 ± 13] / 2. x = 8 or x = −5. The length is 8 m and the width is 5 m. −5 m cannot be a length. Check: 8 × 5 = 40 m².
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BSEB model paper · practice
0/6This model set is for practice. It is not a question from any year’s annual examination.
1
Which of these has D = 0?
💡 Hint:
A perfect square (x − 3)².
Explanation:
D = 36 − 36 = 0.
2
The sum of the roots of x² − 5x + 6 = 0 is —
💡 Hint:
2 + 3.
Explanation:
The roots are 2 and 3, sum 5, which is −b/a.
3
If the roots are equal, D = ______.
💡 Hint:
0.
Explanation:
The square root is zero.
4
A student wrote only 2 as the root of x² − 4 = 0. What was missed?
Model answer (board format)
The ± was missed. From x² = 4, x = 2 or x = −2. Both are exact roots.
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5
Solve 2x² + 3x − 2 = 0 by the formula. Write D and both roots as fractions.
Model answer (board format)
a = 2, b = 3, c = −2. D = 9 + 16 = 25. x = [−3 ± 5] / 4. x = 2/4 = 1/2 or x = −8/4 = −2.
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6
Assertion (A): x² + 4 = 0 has no real root.
Reason (R): Its D is 16.
Explanation:
A is true, x² = −4. R is false. In standard form x² + 0x + 4 = 0, D = −16.
CBSE-style questions
0/6These are competency-based practice questions. They are not a copy of any year’s paper.
1
A student says x² + 4x + 4 = 0 has a single root, so the nature is “no root”. The correct statement is —
💡 Hint:
D = 16 − 16 = 0.
Explanation:
D = 0, both roots are −2. “No root” is the case D < 0.
2
A room is 2 m longer than it is wide, and the area is 48 m². The width is —
💡 Hint:
w(w + 2) = 48.
Explanation:
w² + 2w − 48 = 0, (w + 8)(w − 6) = 0. The width is 6 m and the length is 8 m.
3
Assertion (A): 5x² − 20 = 0 has real roots.
Reason (R): D = 0 − 4·5·(−20) = 400 > 0.
Explanation:
Both are true. The roots are x = ±2.
4
Assertion (A): A square is completed by adding the number only on the left.
Reason (R): The equation must stay balanced, so the same number is added on the right too.
Explanation:
A is false. R is true.
5
A notebook writes D = b² + 4ac. For x² − 4x + 3 = 0, what D does this mistake give, and what is the correct D?
Model answer (board format)
The wrong arithmetic gives 16 + 12 = 28. The correct formula is D = b² − 4ac = 16 − 12 = 4. The roots are 3 and 1, not those of 28.
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6
What do exercises 4.2 and 4.3 of the current book ask? Where is completing the square in the older book?
Model answer (board format)
In the current NCERT, 4.2 is factorisation and 4.3 is the nature of the roots and the formula. Completing the square is exercise 4.3 of the older Bihar book.
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🏛️ Board exam corner — Bihar Board (BSEB)— CBSE
Switch board with BSEB | CBSE above. The lessons follow the same NCERT chapter.
BSEB model · practiceBoard tip
This page has no verified annual-exam question, because no source page has been added. The model set below is practice in the board pattern.
🏛️ Model questions on one page →
The verified label will be used only when a source page for the question is available.
CBSE-style · competency-based
These are case and assertion-reason practice items. Do not treat them as past CBSE questions.
🔁 Spaced review — today’s questions
Wrong questions return soon; correct ones return after a few days.
🧠 What you learned + equation sheet
What you learned
- The standard form is ax² + bx + c = 0, and a is not zero.
- In factorisation, set each linear factor equal to zero on its own.
- In completing the square, add (b/2)² on both sides.
- The formula is x = [−b ± √D] / (2a), where D = b² − 4ac.
- D > 0 means two different roots, D = 0 two equal roots, D < 0 no real root.
- In a word problem, drop a root that breaks the condition, such as a negative length.
| What | Keep this |
|---|---|
| Standard form | ax² + bx + c = 0, a ≠ 0 |
| Discriminant | D = b² − 4ac |
| Two different roots | D > 0 |
| Two equal roots | D = 0, x = −b/(2a) |
| No real root | D < 0 |
| Formula | x = [−b ± √D] / (2a) |
The notes are original writing. The textbook was used only for activity order and numbers. “Verified” will be used only on a question that has a source page.