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NCERT Class X Chapter 4: Quadratic Equations Formulas, Tricks & Tips Here are the most important formulas, tips and exam strategies to solve Quadratic Equations effectively. Standard Form of Quadratic Equation A quadratic equation is written as: \( ax^2 + bx + c = 0, \; a \neq 0 \) Formulas Discriminant: \( \Delta = b^2 - 4ac \) Quadratic Formula (Roots): \( x = \frac{-b \pm \sqrt{\Delta}}{2a} \) Nature of Roots: \( \Delta > 0 \Rightarrow \) Two distinct real roots \( \Delta = 0 \Rightarrow \) Two equal real roots \( \Delta < 0 \Rightarrow \) No real roots Relation between roots and coefficients: If roots are \( \alpha \) and \( \beta \): \( \alpha + \beta = -\frac{b}{a}, \quad \alpha \beta = \frac{c}{a} \) Factorization Method (Splitting the Middle Term) For a quadratic equation: \( ax^2 + bx + c = 0 \) We find two numbers \(m\) and \(n\) such that: \( m \times n = a \times c \)    and    \( m + n = b \) Then we can ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 5 Question: Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2? If so, find its length and breadth. Given: A rectangular park with Perimeter (P) = 80 m Area (A) = 400 m 2 To Find: 1. Is it possible to design such a rectangular park? 2. If yes, find its length and breadth. Formula: Perimeter of a rectangle is given by: P = 2(Length + Breadth) Area of a rectangle is given by: A = Length × Breadth For a quadratic equation of the form ax 2 + bx + c = 0, real solutions exist if and only if its discriminant (Δ) is greater than or equal to zero. Δ = b 2 - 4ac ≥ 0 Solution: Let the length of the rectangular park be 'l' meters Let the breadth of the rectangular park be 'b' meters. Given Perimeter (P) = 80 m WKT, P = 2(l + b) ⇒  2(l + b) = 80 ⇒  l + b = 40 (E...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 4 Question: Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48. Given: The sum of the present ages of two friends is 20 years. Four years ago, the product of their ages was 48. To Find: 1. Is the given situation possible? 2. If yes, determine their present ages. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, real solutions exist if and only if its discriminant (Δ) is greater than or equal to zero. Δ = b 2 - 4ac ≥ 0 Solution: Let, The present age of the first friend (in years) = x Since the sum of their ages is 20 years, The present age of the second friend(in years) = 20 - x Four years ago: Age of the first friend (in years) = x - 4 Age of the second friend (in years...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 Question 3 Question: Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m^2? If so, find its length and breadth. Given: A rectangular mango grove of Length (L) is twice its breadth (B) ⇒ L = 2B Area (A) = 800 m 2 To Find: 1. Is it possible to design such a rectangular mango grove? 2. If yes, find its length and breadth. Formula: Area of a rectangle is given by: Area = Length × Breadth (A = L × B) A quadratic equation of the form ax 2 + bx + c = 0, for real solutions to exist, the discriminant (Δ) must be greater than or equal to zero. Δ = b 2 - 4ac ≥ 0 Solution: Let the breadth of the rectangular mango grove be 'x' meters. According to the problem, the length is twice its breadth: Length = 2x meters. The area of the rectangular mango grove is given as 800 m 2 . WKT,...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 2 (ii) Question: Find the values of k for each of the following quadratic equations, so that they have two equal roots : kx (x – 2) + 6 = 0. Given: The equation: kx (x – 2) + 6 = 0 To Find: The value(s) of 'k' such that the equation has two equal roots. Formula: A quadratic equation of the form ax 2 + bx + c = 0 to have two equal roots, its discriminant (Δ) must be equal to zero. Δ = b 2 - 4ac = 0 Solution: The given equation is kx (x – 2) + 6 = 0. First, rewrite the equation in the standard quadratic form ax 2 + bx + c = 0: ⇒ kx 2 - 2kx + 6 = 0 Comparing it with the standard form ax 2 + bx + c = 0, we have: a = k b = -2k c = 6 For the equation to be a quadratic equation, the coefficient 'a' cannot be zero, i.e., k ≠ 0. (Condition 1) For two equal roots, the discriminant must be zero (Δ = 0): ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 2 (i) Question: Find the values of k for each of the following quadratic equations, so that they have two equal roots: 2x 2 + kx + 3 = 0. Given: The quadratic equation: 2x 2 + kx + 3 = 0 To Find: The value(s) of 'k' such that the equation has two equal roots. Formula: For quadratic equation of the form ax 2 + bx + c = 0 to have two equal roots, its discriminant (Δ) must be equal to zero. Δ = b 2 - 4ac = 0 Solution: The given quadratic equation is 2x 2 + kx + 3 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2 b = k c = 3 For two equal roots, the discriminant must be zero (Δ = 0): ⇒ b 2 - 4ac = 0 ⇒ (k) 2 - 4(2)(3) = 0 ⇒ k 2 - 24 = 0 ⇒ k 2 = 24 ⇒ k = ±√24 ⇒ k = ±√(4 × 6) ⇒ k = ±2√6 Result: The values of k for which the quadratic equation 2x 2 + kx + ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 1 (iii) Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : 2x 2 – 6x + 3 = 0 Given: The quadratic equation: 2x 2 – 6x + 3 = 0 To Find: The nature of the roots. If real roots exist, find them. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots. WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 2x 2 – 6x + 3 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2,...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 1 (ii) Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : 3x 2 – 4√3x + 4 = 0. Given: The quadratic equation: 3x 2 – 4√3x + 4 = 0 To Find: The nature of the roots. If real roots exist, find them. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots. WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 3x 2 – 4√3x + 4 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.3 1 (i) Question: Find the nature of the roots of the following quadratic equations. If the real roots exist, find them : 2x 2 – 3x + 5 = 0. Given: The quadratic equation: 2x 2 – 3x + 5 = 0 To Find: The nature of the roots. If real roots exist, find them. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots. WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 2x 2 – 3x + 5 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2 ...

NCERT Class X Chapter 4: Quadratic Equation Example 9 Question: Find the discriminant of the equation 3x 2 – 2x + (1/3) = 0 and hence find the nature of its roots. Find them, if they are real.. Given: The quadratic equation: 3x 2 – 2x + 1 3 = 0 To Find: 1. The discriminant of the equation. 2. The nature of its roots. 3. The roots, if they are real. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots (or two distinct complex roots). WKT, if the roots are real, they can be found using the quadratic formula: x = -b ± √Δ 2a Solution: The given quadratic equation is 3x 2 – 2x ...

NCERT Class X Chapter 4: Quadratic Equation Example 8 Question: A pole has to be erected at a point on the boundary of a circular park of diameter 13 metres in such a way that the differences of its distances from two diametrically opposite fixed gates A and B on the boundary is 7 metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?. Given: A circular park with diameter AB = 13 metres. A pole (P) is erected on the boundary. The difference of its distances from two diametrically opposite fixed gates A and B is 7 metres.                                 To Find: 1. Is it possible to erect the pole under these conditions? 2. If yes, at what distances from the two gates should the pole be erected? Formula: WKT, an angle inscribed ...

NCERT Class X Chapter 4: Quadratic Equation Example 7 Question: Find the discriminant of the quadratic equation 2x 2 – 4x + 3 = 0, and hence find the nature of its roots. Given: The quadratic equation: 2x 2 – 4x + 3 = 0 To Find: 1. The discriminant of the equation. 2. The nature of its roots. Formula: WKT, for a quadratic equation of the form ax 2 + bx + c = 0, the discriminant (Δ) is given by: Δ = b 2 - 4ac WKT, the nature of roots is determined by the value of the discriminant (Δ): • If Δ > 0, there are two distinct real roots. • If Δ = 0, there are two equal real roots. • If Δ < 0, there are no real roots (or two distinct complex roots). Solution: The given quadratic equation is 2x 2 – 4x + 3 = 0. Comparing it with the standard form ax 2 + bx + c = 0, we have: a = 2 b = -4 c = 3 Now, calculate the discriminant (Δ): ⇒ Δ = b 2 - 4ac ⇒ Δ = (-4)...

NCERT Class X Chapter 4:Quadratic Equation Exercise 4.2 Question 2 Question: Solve the problems given in Example 1. Solution: Example 1 subdivision i solution Example 1 subdivision ii solution Next Question Solution: NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 Question 3. Explore more in Quadratic Equations chapter: Click this link to explore more NCERT Class X Chapter 4 Quadratic Equations solutions Explore more: Click this link to explore more NCERT Class X chapter solutions © Kaliyuga Ekalavya. All rights reserved.

NCERT Class X Chapter 4:Quadratic Equation Exercise 4.2 Question 3 Question: Find two numbers whose sum is 27 and product is 182. Given: Sum of two numbers = 27 Product of two numbers = 182 To Find: The two numbers. Formula: For a quadratic equation ax² + bx + c = 0 , we can find two numbers m and n such that:                               ax² + bx + c = ax² + mx + nx + c where m × n = a × c m + n = b Cases: If (a × c > 0) and b < 0 , both m and n are negative . If (a × c > 0) and b > 0 , both m and n are positive . If (a × c < 0) , the numbers m and n have opposite signs . Solution: Let one of the numbers be 'x' and the other number be 'y' ⇒x + y = 27 ⇒y = 27 - x According to the given condition, their product is ...

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.2 4 Question: Find two consecutive positive integers, sum of whose squares is 365. Given: Two consecutive positive integers. Sum of their squares = 365. To Find: The two consecutive positive integers. Formula: WKT, if the first integer is x, the next consecutive integer is (x + 1). WKT, the algebraic identity (a + b) 2 = a 2 + 2ab + b 2 . WKT, For a quadratic equation ax² + bx + c = 0 , we can find two numbers m and n such that:                               ax² + bx + c = ax² + mx + nx + c where m × n = a × c m + n = b Cases: If (a × c > 0) and b < 0 , both m and n are negative . If (a × c > 0) and b > 0 , both m and n are positive . If (a × c < 0) , the numbers m and n have opposite signs ....