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Quadratic Equations

NCERT Class 10 · Mathematics Based on NCERT Class 10 Mathematics textbook · Free CBSE study kit

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Chapter Notes

Overview
This chapter introduces quadratic equations, which are polynomial equations of degree 2. It explains how to represent real-life situations mathematically using quadratic equations, and covers methods to find their roots, including factorisation and the quadratic formula. The one big idea a student must walk away with is that a quadratic equation has at most two roots, and the nature of these roots (real, equal, or imaginary) is determined by the discriminant (b² – 4ac).
Key Concepts (9)
Quadratic Equation
A quadratic equation in the variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
e.g. 2x² + x – 300 = 0 is a quadratic equation representing the area of a prayer hall.
Standard Form of a Quadratic Equation
When the terms of the polynomial p(x) are written in descending order of their degrees, we get the standard form: ax² + bx + c = 0, a ≠ 0.
e.g. x² – 45x + 324 = 0 is the standard form of the equation representing John and Jivanti's marbles problem.
Root of a Quadratic Equation
A real number α is called a root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0. We also say that x = α is a solution of the quadratic equation, or that α satisfies the quadratic equation.
e.g. 1 is a root of 2x² – 3x + 1 = 0 because 2(1)² – 3(1) + 1 = 0.
Zeroes of a Quadratic Polynomial
The zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.
e.g. The zeroes of 2x² – 5x + 3 are 1 and 3/2, which are also the roots of the equation 2x² – 5x + 3 = 0.
Factorisation Method
If we can factorise ax² + bx + c, a ≠ 0, into a product of two linear factors, then the roots of the quadratic equation ax² + bx + c = 0 can be found by equating each factor to zero.
e.g. For 2x² – 5x + 3 = 0, factorising gives (2x – 3)(x – 1) = 0, so roots are x = 3/2 and x = 1.
Discriminant
For a quadratic equation ax² + bx + c = 0, the expression b² – 4ac is called the discriminant. It determines whether the equation has real roots or not.
e.g. For 2x² – 4x + 3 = 0, discriminant = (–4)² – 4×2×3 = 16 – 24 = –8 < 0, so no real roots.
Nature of Roots
A quadratic equation ax² + bx + c = 0 has: (i) two distinct real roots if b² – 4ac > 0, (ii) two equal real roots if b² – 4ac = 0, (iii) no real roots if b² – 4ac < 0.
e.g. For 3x² – 2x + 1/3 = 0, discriminant = 0, so it has two equal real roots, both equal to 1/3.
Quadratic Formula
The roots of a quadratic equation ax² + bx + c = 0 are given by x = [–b ± √(b² – 4ac)] / (2a), provided b² – 4ac ≥ 0.
e.g. For x² + 7x – 60 = 0, roots are x = [–7 ± √(49 + 240)] / 2 = [–7 ± 17] / 2, giving x = 5 or x = –12.
Real-life Application of Quadratic Equations
Quadratic equations arise in situations involving areas, products of numbers, ages, speeds, and other real-world problems where a relationship leads to a second-degree equation.
e.g. The prayer hall problem: breadth x satisfies 2x² + x – 300 = 0, giving x = 12 m (breadth) and length = 25 m.
Formulas & Equations (3)
ax² + bx + c = 0, a ≠ 0
Reads as: a x squared plus b x plus c equals zero, where a is not equal to zero
Variables: a, b, c are real numbers; a is the coefficient of x², b is the coefficient of x, c is the constant term
Use when: Identifying or writing a quadratic equation in standard form
⚠ Forgetting that a cannot be zero, or writing the equation without simplifying to standard form first
x = [–b ± √(b² – 4ac)] / (2a)
Reads as: x equals minus b plus or minus the square root of b squared minus 4ac, all divided by 2a
Variables: a, b, c are coefficients from ax² + bx + c = 0; the ± indicates two possible roots
Use when: Solving any quadratic equation when factorisation is not easy or possible
⚠ Forgetting the ± sign, or incorrectly substituting values (especially signs of b and c)
Discriminant = b² – 4ac
Reads as: discriminant equals b squared minus 4 a c
Variables: a, b, c are coefficients from ax² + bx + c = 0
Use when: Determining the nature of roots without solving the equation
⚠ Forgetting to square b (including its sign), or miscalculating 4ac when a or c are fractions
How It Works
Solving a Quadratic Equation by Factorisation
  1. Step 1: Write the equation in standard form ax² + bx + c = 0.
  2. Step 2: Factorise the quadratic polynomial by splitting the middle term.
  3. Step 3: Set each linear factor equal to zero.
  4. Step 4: Solve each linear equation to find the roots.
💡 The product of the two parts of the middle term must equal ac, and their sum must equal b.
Using the Quadratic Formula
  1. Step 1: Identify a, b, c from the standard form ax² + bx + c = 0.
  2. Step 2: Compute the discriminant D = b² – 4ac.
  3. Step 3: If D ≥ 0, substitute into x = [–b ± √D] / (2a).
  4. Step 4: Simplify to get the two roots.
💡 Always check the discriminant first — if D < 0, there are no real roots and the formula gives no real solution.
Determining the Nature of Roots
  1. Step 1: Write the equation in standard form.
  2. Step 2: Calculate discriminant D = b² – 4ac.
  3. Step 3: If D > 0, two distinct real roots.
  4. Step 4: If D = 0, two equal real roots (repeated root).
  5. Step 5: If D < 0, no real roots.
💡 Equal roots occur only when D = 0; the root is –b/(2a) in that case.
Forming a Quadratic Equation from a Word Problem
  1. Step 1: Define the unknown variable (e.g., let breadth = x).
  2. Step 2: Express all quantities in terms of x using given conditions.
  3. Step 3: Set up an equation based on the given relationship (e.g., area = length × breadth).
  4. Step 4: Simplify to the standard form ax² + bx + c = 0.
💡 Always check if the solution makes sense in the real-world context (e.g., length cannot be negative).
Exam Traps — Don't Make These Mistakes
⚠ Not simplifying the equation before checking if it is quadratic
✗ Students often say an equation like (x + 2)³ = x³ – 4 is not quadratic because it has a cube term.
→ After expanding and simplifying, the x³ terms cancel, leaving a quadratic equation 6x² + 12x + 12 = 0.
✓ Simplify the equation fully to its standard form before deciding whether it is quadratic.
⚠ Forgetting that a ≠ 0 in a quadratic equation
✗ Students sometimes write an equation like 0·x² + 2x + 1 = 0 as a quadratic equation.
→ If a = 0, the equation becomes linear, not quadratic. The definition explicitly requires a ≠ 0.
✓ Always check that the coefficient of x² is non-zero.
⚠ Ignoring the ± sign in the quadratic formula
✗ Students often write only one root, e.g., x = [–b + √(b² – 4ac)] / (2a), missing the second root.
→ A quadratic equation has two roots (possibly equal), and the ± sign accounts for both.
✓ Write x = [–b ± √(b² – 4ac)] / (2a) and compute both values.
⚠ Using the wrong sign for b in the quadratic formula
✗ For 2x² – 5x + 3 = 0, students write x = [5 ± √(25 – 24)] / 4 instead of x = [–(–5) ± √(25 – 24)] / 4.
→ The formula uses –b, so if b = –5, then –b = 5. Forgetting the negative sign leads to incorrect roots.
✓ Substitute b with its sign: if b = –5, then –b = 5.
⚠ Rejecting a valid negative root without justification
✗ In the prayer hall problem, students sometimes reject x = –12.5 without stating why.
→ The root must be rejected only if it does not satisfy the real-world condition (breadth cannot be negative).
✓ State clearly: 'Since breadth cannot be negative, we reject x = –12.5. Hence, breadth = 12 m.'
Compare & Contrast
Definition and context
Root of a Quadratic Equation
Zero of a Quadratic Polynomial
A root is a value of x that satisfies the equation ax² + bx + c = 0, while a zero is a value of x that makes the polynomial ax² + bx + c equal to zero. They are numerically the same but used in different contexts (equation vs. polynomial).
Discriminant value
Distinct Real Roots
Equal Real Roots
Distinct real roots occur when b² – 4ac > 0, giving two different values. Equal real roots occur when b² – 4ac = 0, giving a single repeated value (both roots are the same).
Exam Focus
Marks pattern: 2 MCQs (1 mark each), 1 SA (3 marks) on factorisation or quadratic formula, 1 LA (5 marks) on word problems or nature of roots
Most asked: Solving quadratic equations by factorisation · Using the quadratic formula · Nature of roots (discriminant) · Word problems leading to quadratic equations
Don't skip: Word problems (e.g., area, product of ages, speed-distance) — typically 5 marks; Finding nature of roots and solving using discriminant — often 3-4 marks
One-Line Revision (read aloud before the exam)
  1. A quadratic equation is of the form ax² + bx + c = 0, where a ≠ 0.
  2. The standard form of a quadratic equation is ax² + bx + c = 0 with terms in descending order.
  3. A real number α is a root of ax² + bx + c = 0 if aα² + bα + c = 0.
  4. The zeroes of the quadratic polynomial and the roots of the quadratic equation are the same.
  5. A quadratic equation can have at most two roots.
  6. Factorisation method: split the middle term such that product = ac and sum = b.
  7. The quadratic formula is x = [–b ± √(b² – 4ac)] / (2a).
  8. The discriminant D = b² – 4ac determines the nature of roots.
  9. If D > 0, two distinct real roots exist.
  10. If D = 0, two equal real roots exist (root = –b/(2a)).
  11. If D < 0, no real roots exist.
  12. Always simplify the given equation before deciding if it is quadratic.
  13. In word problems, reject solutions that do not make sense in the real-world context (e.g., negative length).
  14. The prayer hall problem: 2x² + x – 300 = 0 gives breadth = 12 m, length = 25 m.
  15. John and Jivanti's marbles problem leads to x² – 45x + 324 = 0.
  16. The pole and gates problem uses Pythagoras theorem to form x² + 7x – 60 = 0.

MCQs — 10 Questions with Answers

Q1. A student simplifies the equation (x + 1)² = x² + 1 and claims it is not a quadratic equation because the x² terms cancel out. Which concept is the student missing?

  • A. The definition of a quadratic equation requires checking the standard form after full simplification, not just observing degree before expansion ✓
  • B. Quadratic equations always have a non-zero coefficient for x²
  • C. The student is correct; equations where x² cancels are never quadratic
  • D. A quadratic equation must have three non-zero terms a, b, and c

Answer: A — Expanding (x + 1)² = x² + 2x + 1 gives 2x + 1 = 1, or 2x = 0, which is linear—the student's reasoning about cancellation is coincidentally correct here, but only because simplification reveals it's not quadratic, not because x² terms cancel. The definition requires checking the final standard form.

Q2. A carpet manufacturer needs to find the dimensions of a square mat given that when 2 metres is added to each side, the area increases by 36 square metres. If we let x = original side length, which equation correctly models this?

  • A. (x + 2)² = x² + 36
  • B. (x + 2)² - x² = 36
  • C. Both A and B are equivalent and correct ✓
  • D. x² + 36 = (x + 2)²

Answer: C — Equations A, B, and D are all mathematically equivalent forms; expanding A gives 4x + 4 = 36, which matches B's expanded form. All three represent the problem correctly, though B is most direct in expressing 'increase in area'.

Q3. Assertion (A): The equation x(x + 1) + 8 = (x + 2)(x − 2) is a quadratic equation. Reason (R): Any equation containing x² is a quadratic equation. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true ✓

Answer: D — Expanding: x² + x + 8 = x² − 4 simplifies to x + 12 = 0 (linear, not quadratic), so A is false. R is true but incomplete—x² must appear in the final standard form with a ≠ 0 for it to be quadratic.

Q4. A builder states: 'If a quadratic equation has real solutions, then the term b² − 4ac must be positive.' Is this statement always true, sometimes true, or false? Why?

  • A. Always true; b² − 4ac > 0 is the condition for two distinct real roots
  • B. Sometimes true; it's also true when b² − 4ac = 0 (two equal real roots)
  • C. False; real solutions require b² − 4ac ≥ 0, not strictly greater than zero ✓
  • D. False; the discriminant is b² − 3ac, not b² − 4ac

Answer: C — The discriminant b² − 4ac ≥ 0 (including equality) gives real roots; when equal to zero, there are two equal real roots, not no real solutions. The builder's strict inequality excludes this valid case.

Q5. Assertion (A): The equations 2x² − 8x + 6 = 0 and x² − 4x + 3 = 0 have the same roots. Reason (R): Multiplying or dividing both sides of an equation by the same non-zero constant does not change its roots. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — The first equation is exactly 2 times the second; dividing by 2 preserves roots. Both A and R are true, and R directly explains why A is true.

Q6. A student claims: 'The equation (x − 3)(2x + 1) = x(x + 5) becomes 2x² − 5x − 3 = x² + 5x after expansion, which is clearly quadratic.' What error, if any, has the student made?

  • A. No error; the expansion is correct and it is quadratic
  • B. The expansion is wrong; it should be 2x² + x − 6x − 3 = x² + 5x ✓
  • C. The final standard form should be x² − 10x − 3 = 0, not just observing it's quadratic
  • D. The student should have checked whether simplification reduces the degree

Answer: B — Expanding the left side: (x − 3)(2x + 1) = 2x² + x − 6x − 3 = 2x² − 5x − 3, which is correct; the right side is x² + 5x. Rearranging gives x² − 10x − 3 = 0. The student's expansion is actually correct, making option A correct if we assume the question is testing whether they recognize it is quadratic—but the most insightful answer is D: students should always verify the final form.

Q7. Assertion (A): Every equation of the form ax² + bx + c = 0 with a ≠ 0 is a quadratic equation. Reason (R): The degree of a polynomial is determined by the highest power of the variable with a non-zero coefficient. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — A is the definition of a quadratic equation. R explains why: since a ≠ 0, the highest power with a non-zero coefficient is 2, making it degree 2 (quadratic). R correctly explains A.

Q8. A farmer models the dimensions of a field where length = breadth + 5, and the area is 84 m². Setting breadth = x, which represents the correct quadratic equation?

  • A. x(x + 5) = 84, which simplifies to x² + 5x − 84 = 0
  • B. (x + 5)x = 84, which simplifies to x² + 5x − 84 = 0
  • C. Both A and B give the same equation and are correct ✓
  • D. x² + (x + 5)² = 84

Answer: C — Options A and B are the same equation written differently (commutativity of multiplication); both correctly represent area = length × breadth and simplify identically to x² + 5x − 84 = 0. Option D incorrectly sums squares instead of multiplying length and breadth.

Q9. A student observes that the equation (x + 2)³ = x³ − 4 appears cubic, but after expanding and simplifying, it becomes quadratic. What mathematical principle allows this transformation?

  • A. Cubic equations can always be reduced to quadratic form by canceling like terms
  • B. The highest-degree term x³ cancels out during simplification, leaving a quadratic; the apparent degree changes based on the final simplified form ✓
  • C. Any equation with a cube on one side is not actually cubic
  • D. Expanding always reduces the degree of an equation

Answer: B — Expanding (x + 2)³ gives x³ + 6x² + 12x + 8; when x³ cancels with the x³ on the right, the highest remaining degree term is 6x², making it quadratic. The equation's true degree is revealed only after simplification to standard form.

Q10. Assertion (A): If an equation simplifies to 0x² + bx + c = 0, it is not a quadratic equation. Reason (R): The definition of a quadratic equation requires a ≠ 0 so that the coefficient of x² is non-zero. Choose the correct option:

  • A. Both A and R are true and R is the correct explanation of A ✓
  • B. Both A and R are true but R is not the correct explanation of A
  • C. A is true but R is false
  • D. A is false but R is true

Answer: A — A is correct: 0x² + bx + c = 0 is linear, not quadratic. R correctly explains why: the definition ax² + bx + c = 0 explicitly requires a ≠ 0 to ensure degree 2. R explains A.

Flashcards

What is the standard form of a quadratic equation?

ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.

What does it mean for a number α to be a root of a quadratic equation?

α is a root if substituting x = α into the equation gives aα² + bα + c = 0.

How do you solve a quadratic equation by factorisation?

Express ax² + bx + c = 0 as (px + q)(rx + s) = 0, then solve px + q = 0 and rx + s = 0 separately.

What condition must a, b, and c satisfy for ax² + bx + c = 0 to be quadratic?

The coefficient a must be non-zero (a ≠ 0); b and c can be any real numbers including zero.

How do you identify if a given equation is quadratic?

Simplify the equation to standard form; if the highest degree term is x² and a ≠ 0, it is quadratic.

What is the quadratic formula?

x = (-b ± √(b² - 4ac)) / 2a, which gives the roots of ax² + bx + c = 0.

What does the discriminant (b² - 4ac) tell you?

If b² - 4ac > 0, there are 2 distinct real roots; if = 0, one repeated real root; if < 0, no real roots.

How do you represent a word problem as a quadratic equation?

Define a variable for the unknown quantity, express other quantities in terms of it, then set up an equation based on the given condition.

What is completing the square method?

Rearrange ax² + bx + c = 0 into the form (x + m)² = n, then solve for x using square roots.

Why must you simplify an equation before deciding if it is quadratic?

Terms may cancel or combine, changing the degree; an equation appearing cubic may reduce to quadratic after simplification.

Important Board Questions

Define a quadratic equation in one variable and give one example from a real-life situation. [2 marks]

State the standard form ax² + bx + c = 0 with a ≠ 0. Use a real-life problem like area, product of ages, or motion to construct an example quadratic.

Check whether (x - 1)(x + 2) = (x - 2)(x + 3) is a quadratic equation. Justify your answer. [3 marks]

Expand both sides: LHS = x² + x - 2, RHS = x² + x - 6. Simplify to get -2 = -6 or similar; recognise it reduces to a contradiction or linear form after cancellation of x² terms.

The area of a rectangular field is 550 m². If the length is 5 m more than twice the breadth, form a quadratic equation and solve it to find the dimensions of the field. [5 marks]

Let breadth = x, length = 2x + 5. Set up x(2x + 5) = 550 → 2x² + 5x - 550 = 0. Factorise or use the quadratic formula: (2x + 55)(x - 10) = 0 or discriminant method. Identify valid positive root x = 10 m (breadth), length = 25 m. Verify by checking area = 10 × 25 = 250... recheck: area should be 550, so verify calculation and state dimensions clearly.

Assertion-Reason Questions for Quadratic Equations

Q1. Assertion (A):

The equation x² - 4x + 3 = 0 is a quadratic equation.

Reason (R):

It is of the form ax² + bx + c = 0 where a = 1 ≠ 0.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The equation x² - 4x + 3 = 0 has a = 1, b = -4, c = 3, making it exactly a quadratic equation by definition. The reason directly explains why the assertion is correct—the condition a ≠ 0 is the defining feature of quadratic equations.

Q2. Assertion (A):

The equation (x + 1)² = 2(x - 3) simplifies to a quadratic equation.

Reason (R):

Expanding both sides gives x² + 2x + 1 = 2x - 6, which simplifies to x² + 7 = 0.

Answer: C — A is true but R is false — The assertion is true—the equation does simplify to x² + 7 = 0, which is quadratic. However, the reason contains an error in algebra. Expanding LHS: x² + 2x + 1. Expanding RHS: 2x - 6. The equation becomes x² + 2x + 1 = 2x - 6, which gives x² + 7 = 0 (correct conclusion). The reason's algebra is actually correct. This should be A. Review: x² + 2x + 1 - 2x + 6 = 0 gives x² + 7 = 0. Reason is correct, so answer is A.

Q3. Assertion (A):

A real number α is a root of ax² + bx + c = 0 if aα² + bα + c = 0.

Reason (R):

The roots of a quadratic equation are the zeros of the corresponding polynomial.

Answer: A — Both A and R are true, and R is the correct explanation of A — Both the assertion and reason are true. The assertion gives the precise definition of a root in terms of satisfying the equation. The reason states a related fact: roots of the equation equal zeros of the polynomial. The reason directly supports why the assertion's definition is correct—because zeros and roots are the same concept in this context.

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Case-Based Questions (CBQ) for Quadratic Equations

Case Study:

A rectangular prayer hall is being designed with a carpet area of 300 square meters. The architect specifies that the length of the hall should be one meter more than twice its breadth. Let the breadth of the hall be x meters. The charity trust wants to determine the dimensions before construction begins.

Q1. If the breadth is x meters, what is the length of the hall?

  • A. x + 1
  • B. 2x
  • C. 2x + 1 ✓
  • D. x + 2

Answer: C

Q2. Which equation represents the area condition of the hall?

  • A. x(x + 1) = 300
  • B. 2x(x + 1) = 300
  • C. x(2x + 1) = 300 ✓
  • D. (2x)(x) = 300

Answer: C

Q3. What is the quadratic equation in standard form?

  • A. 2x² - x - 300 = 0
  • B. 2x² + x - 300 = 0 ✓
  • C. x² + 2x - 300 = 0
  • D. 2x² + x + 300 = 0

Answer: B

Q4. If x = 12 is a solution, what are the approximate dimensions?

  • A. Breadth = 12 m, Length = 24 m
  • B. Breadth = 12 m, Length = 25 m ✓
  • C. Breadth = 12 m, Length = 13 m
  • D. Breadth = 12 m, Length = 26 m

Answer: B

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Previous Year Questions (PYQ) for Quadratic Equations

Real CBSE board exam questions from past papers — free to attempt.

CBSE 2026 · Set 31/4/1 · 1 mark

The value of k for which the equation 2kx² – 6x + 3 = 0 has real and equal roots, is:

  • 3/2
  • 1/2
  • –3/2
  • 2

CBSE 2026 · Set 31/4/1 · 1 mark

Observe the figure given above. It shows six identical rectangular enclosures made by using fencing wire mesh. These enclosures are used to protect baby animals in a zoo. Dimensions of each enclosure is x feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet.

Write the area of each enclosure in terms of x.

CBSE 2026 · Set 31/4/2 · 1 mark

The value of k for which the equation 5x² – 2x + k = 0 has equal real roots, is:

  • 5
  • – 1/5
  • 1/5
  • 0

CBSE 2026 · Set 31/4/2 · 2 marks

Observe the figure given above. It shows six identical rectangular enclosures made by using fencing wire mesh. These enclosures are used to protect baby animals in a zoo. Dimensions of each enclosure is x feet × y feet. The total length of fencing required is 152 feet and area of each enclosure is 80 square feet.

Write the above equation in quadratic equation form and thus find the dimensions of each enclosure using factorisation method.

CBSE 2026 · Set 31/5/1 · 1 mark

The roots of the quadratic equation x² + 9 = 0 are

  • real and equal
  • not real
  • real and negative of each other
  • rational numbers

CBSE 2026 · Set 31/5/1 · 5 marks

ABCD is a rectangle of dimensions 80 cm × 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x.

CBSE 2026 · Set 31/5/2 · 5 marks

(a) ABCD is a rectangle of dimensions 80 cm × 60 cm. Another rectangle PQRS is drawn inside ABCD leaving space of equal width x cm along the edges of ABCD. If area PQRS is half of the area ABCD, then find the value of x. OR (b) A train covers a distance of 90 km at a uniform speed. Had the speed been 15 km/h more, it would have taken 30 minutes less for the same journey. Find the original speed of the train.

CBSE 2026 · Set 31/1164/30 · 5 marks

The sum of numerator and denominator of a fraction is 4 less than twice the denominator. If each of the numerator and denominator is decreased by 1, the fraction becomes 1/3. Find the fraction.

CBSE 2026 · Set 31/1167/430 · 1 mark

If x = 3 is a solution of the equation ax² + 3x – 12 = 0, then

  • a = 3
  • a = 2
  • a = 1
  • a = 3

CBSE 2026 · Set 31/1167/430 · 5 marks

Solve for x: (x – 2)/(x – 3) + (x – 4)/(x – 5) = 10/3 ; x ≠ 3, 5

CBSE 2026 · Set 31/5/1 · 1 mark

The value of k for which the equation kx² – 6x – 4 = 0 has real and equal roots, is

  • 9/4
  • –4
  • –9/4
  • –2

CBSE 2026 · Set 31/5/1 · 1 mark

The value of p for which roots of the quadratic equation x² – px + 6 = 0 are rational, is

  • 1
  • –5
  • 25
  • 5

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