NCERT Class X Chapter 4: Quadratic Equation Example 2 (iv)

NCERT Class X Chapter 4: Quadratic Equation Example 2 (iv)

Question:

Check whether the following are quadratic equations: 

(x + 2)3 = x3 – 4

Given:

Equation: (x + 2)3 = x3 – 4

To Find:

Whether the given equation is a quadratic equation.

Formula:

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

Solution:

The given equation:(x + 2)3 = x3 – 4
WKT,  (a + b)3 = a3 + b3 + 3ab(a + b)
Expand the left side using (a + b)3:

Here, a = x and b = 2.

⇒ x3 + 23 + 3(x)(2)(x + 2)

⇒ x3 + 8 + 6x(x + 2)

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

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

⇒ x3 - x3 + 6x2 + 12x + 8 + 4 = 0

⇒ 6x2 + 12x + 12 = 0
The given equation (x + 2)3 = x3 – 4 simplifies to 6x2 + 12x + 12 = 0.
Compare this equation to the standard form ax2 + bx + c = 0:

Here, a = 6, b = 12, c = 12.

Since a (the coefficient of x2) is 6 (which is not equal to 0), this equation is a quadratic equation.

Result:

The given equation (x + 2)3 = x3 – 4 simplifies to 6x2 + 12x + 12 = 0.
6x2 + 12x + 12 = 0 is in the form ax2 + bx + c = 0 with a = 6 ≠ 0.

Therefore, it is a quadratic equation.
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