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

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

Question:

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

Given:

Equation: (x + 2)3 = 2x (x2 – 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 algebraic identity (a + b)3 = a3 + b3 + 3a2b + 3ab2.

Solution:

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

⇒ x3 + 23 + 3(x2)(2) + 3(x)(22)

⇒ x3 + 8 + 6x2 + 12x
Expand the right side:

⇒ 2x(x2) - 2x(1) = 2x3 - 2x
Substitute the expanded forms back into the original equation:

⇒ x3 + 6x2 + 12x + 8 = 2x3 - 2x
Move all terms to one side to get the standard form ax2 + bx + c = 0:

⇒ x3 - 2x3 + 6x2 + 12x + 2x + 8 = 0

⇒ -x3 + 6x2 + 14x + 8 = 0
This equation is a cubic equation, as the highest power of x is 3.

For it to be a quadratic equation, the coefficient of x3 must be 0, but here it is -1.

Therefore, it is not a quadratic equation.

Result:

The given equation (x + 2)3 = 2x (x2 – 1) simplifies to -x3 + 6x2 + 14x + 8 = 0.

Since the highest power of x is 3 (i.e., the coefficient of x3 is -1 ≠ 0), it is not a quadratic equation; it is a cubic equation.
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