NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 Question 1 (v)

NCERT Class X Chapter 4: Quadratic Equation Exercise 4.1 1(v)

Question:

Check whether the following are quadratic equations :
(2x – 1)(x – 3) = (x + 5)(x – 1)

Given:

Equation: (2x – 1)(x – 3) = (x + 5)(x – 1)

To Find:

Whether the given equation is a quadratic equation.

Formula:

WKT, a quadratic equation is of the form ax2 + bx + c = 0, where a ≠ 0.

WKT, the product of two binomials (a + b)(c + d) = ac + ad + bc + bd.

Solution:

Start with the given equation:

(2x – 1)(x – 3) = (x + 5)(x – 1)
Expand the left side:

⇒ 2x(x) + 2x(-3) - 1(x) - 1(-3) = 2x2 - 6x - x + 3
= 2x2 - 7x + 3
Expand the right side:

⇒ x(x) + x(-1) + 5(x) + 5(-1) = x2 - x + 5x - 5 = x2 + 4x - 5
Substitute the expanded forms back into the equation:

⇒ 2x2 - 7x + 3 = x2 + 4x - 5
Move all terms to one side to get the standard form ax2 + bx + c = 0:

⇒ 2x2 - x2 - 7x - 4x + 3 + 5 = 0

⇒ x2 - 11x + 8 = 0
Compare this equation to the standard form ax2 + bx + c = 0:

Here, a = 1, b = -11, c = 8.
Since the coefficient of x2 (a) is 1 (which is not equal to 0), this equation is a quadratic equation.

Result:

The given equation (2x – 1)(x – 3) = (x + 5)(x – 1) simplifies to x2 - 11x + 8 = 0. 
Since the coefficient of x2 is 1 ≠ 0, it is a quadratic equation.
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