Kaliyuga Ekalavya

NCERT Class X Chapter 4: Quadratic Equation Example 2 (iii) Question: Check whether the following are quadratic equations:  x (2x + 3) = x 2 + 1 Given: Equation: x (2x + 3) = x 2 + 1 To Find: Whether the given equation is a quadratic equation. Formula: A quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. Solution: The given equation: x (2x + 3) = x 2 + 1 Expand the left side: ⇒ 2x 2 + 3x Substitute the expanded form back into the equation: ⇒ 2x 2 + 3x = x 2 + 1 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ 2x 2 - x 2 + 3x - 1 = 0 ⇒ x 2 + 3x - 1 = 0 Compare this equation to the standard form ax 2 + bx + c = 0: Here, a = 1, b = 3, and c = -1. Since the a...

NCERT Class X Chapter 4: Quadratic Equation Example 2 (ii) Question: Check whether the following are quadratic equations:  x(x + 1) + 8 = (x + 2) (x – 2) Given: Equation: x(x + 1) + 8 = (x + 2) (x – 2) To Find: Whether the given equation is a quadratic equation . Formula: WKT, a quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. Solution: The given equation: x(x + 1) + 8 = (x + 2) (x – 2) Expand the left side: ⇒ x 2 + x + 8 WKT, the algebraic identity (a + b)(a - b) = a 2 - b 2 Expand the right side using the identity (a + b)(a - b) ⇒ x 2 - 2 2 = x 2 - 4 Substitute the expanded forms back into the equation: ⇒ x 2 + x + 8 = x 2 - 4 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 2 - x 2 + x + 8 + 4 =...

NCERT Class X Chapter 4: Quadratic Equation Example 2 (i) Question: Check whether the following are quadratic equations:  (x – 2) 2 + 1 = 2x – 3 Given: Equation: (x – 2) 2 + 1 = 2x – 3 To Find: Whether the given equation is a quadratic equation . Formula: A quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. Solution: The given equation: (x – 2) 2 + 1 = 2x – 3 Expand (x – 2) 2 using (a - b) 2 formula (a - b) 2 = a 2 - 2ab + b 2 ⇒ x 2 - 2(x)(2) + 2 2 + 1 = 2x – 3 ⇒ x 2 - 4x + 4 + 1 = 2x – 3 Simplify the left side: ⇒ x 2 - 4x + 5 = 2x – 3 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 2 - 4x - 2x + 5 + 3 = 0 ⇒  x 2 - 6x + 8 = 0 Compare this equation to the standard form ax 2 + bx + c = 0: Here, a = 1, b = -6, c = 8. Since a = 1 (which is not equal to 0), the equation...

NCERT Class X Chapter 4: Quadratic Equation Example 2 (iv) Question: Check whether the following are quadratic equations:  (x + 2) 3 = x 3 – 4 Given: Equation: (x + 2) 3 = x 3 – 4 To Find: Whether the given equation is a quadratic equation. Formula: A quadratic equation is of the form ax 2 + bx + c = 0, where a ≠ 0. Solution: The given equation:(x + 2) 3 = x 3 – 4 WKT,  (a + b) 3 = a 3 + b 3 + 3ab(a + b) Expand the left side using (a + b) 3 : Here, a = x and b = 2. ⇒ x 3 + 2 3 + 3(x)(2)(x + 2) ⇒ x 3 + 8 + 6x(x + 2) ⇒ x 3 + 8 + 6x 2 + 12x Substitute the expanded form back into the original equation: ⇒ x 3 + 6x 2 + 12x + 8 = x 3 – 4 Move all terms to one side to get the standard form ax 2 + bx + c = 0: ⇒ x 3 - x 3 + 6x 2 + 12x + 8 + 4 = 0 ⇒ 6x 2 + 12x + 12 = 0 The given equation (x + 2) 3 = x 3 – 4 simplifies to 6x 2 + 12x + 12 = 0. ...

NCERT Class X Chapter 4: Quadratic Equation Example 1 (ii) Question: Represent the following situations mathematically: A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day. On a particular day, the total cost of production was 750. We would like to find out the number of toys produced on that day. Given: Cost of production of each toy = 55 - (number of toys produced) Total cost of production on a particular day = 750 rupees To Find: The number of toys produced on that day. Formula: Total cost of production = (Number of toys) × (Cost of production of each toy) Solution: Let,the number of toys produced in a day = x Then,the cost of production of each toy = 55 - x Total cost of production = (Number of toys) × (Cost of each toy) Total cost of production = x(55...