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

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

Question:

Check whether the following are quadratic equations :
x3 – 4x2 – x + 1 = (x – 2)3

Given:

Equation: x3 – 4x2 – x + 1 = (x – 2)3

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,   (a - b)3 = a3 - b3 - 3a2b + 3ab2.

Solution:

The given equation: x3 – 4x2 – x + 1 = (x – 2)3
Expand the right side using (a - b)3 = a3 - b3 - 3a2b + 3ab2:

Here, a = x and b = 2.
⇒ x3 - 23 - 3(x2)(2) + 3(x)(22)

⇒ x3 - 8 - 6x2 + 12x
Substitute the expanded form back into the original equation:

⇒ x3 – 4x2 – x + 1 = x3 - 6x2 + 12x - 8
Move all terms to one side to get the standard form ax2 + bx + c = 0:

⇒ x3 - x3 - 4x2 + 6x2 - x - 12x + 1 + 8 = 0

⇒ 2x2 - 13x + 9 = 0
Compare this equation to the standard form ax2 + bx + c = 0:

Here, a = 2, b = -13, c = 9.
Since the coefficient of x2 (a) is 2 (which is not equal to 0), this equation is a quadratic equation.

Result:

The given equation x3 – 4x2 – x + 1 = (x – 2)3 simplifies to 2x2 - 13x + 9 = 0. S
Since the coefficient of x2 is 2 ≠ 0, it is a quadratic equation.
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