Which of the following is a quadratic equation
a. x² + 2x + 1 = (4 - x)² + 3
b. -2x² = (5 - x)(2x - 2/5)
c. (k + 1)x² + (3/2)x = 7 where k = -1
d. x³ - x² = (x - 1)³

Solution:

We have to find the quadratic equation.

The standard form of a quadratic equation is ax² + bx + c = 0 in variable x.

Where a, b, and c are real numbers and a ≠ 0.

We need to check if the degree of the given equations is 2.

From the options,

A) x² + 2x + 1 = (4 - x)² + 3

By using algebraic identity,

(a - b)² = a² - 2ab + b²

x² + 2x + 1 = 16 + x² - 8x + 3

By grouping,

x² - x² + 2x + 8x + 1 - 16 - 3 = 0

10x - 18 = 0

The degree of the equation is 1.

Therefore, x² + 2x + 1 = (4 - x)² + 3 is not a quadratic equation.

B) -2x² = (5 - x)(2x - 2/5)

-2x² = (5 - x)(10x - 2)/5

-10x² = (5 - x)(10x - 2)

By multiplicative and distributive property,

-10x² = 50x - 10 - 10x² + 2x

By grouping,

-10x² + 10x² + 52x - 10 = 0

52x - 10 = 0

The degree of the equation is 1.

Therefore, -2x² = (5 - x)(2x - 2/5) is not a quadratic equation.

C) (k + 1)x² + (3/2)x = 7 where k = -1

(-1 + 1)x² + (3/2)x = 7

3/2x = 7

The degree of the equation is 1.

Therefore, (k + 1)x² + (3/2)x = 7 where k = -1 is not a quadratic equation.

D) x³ - x² = (x - 1)³

By using algebraic identity,

(a - b)³ = a³ - b³ - 3a²b + 3ab²

x³ - x² = x³ - 1 - 3x² + 3x

By grouping,

x³ - x³ - x² + 3x² - 3x + 1 = 0

2x² - 3x + 1 = 0

The degree of the equation is 2.

Therefore, x³ - x² = (x - 1)³ is a quadratic equation.

✦ Try This: Which of the following is a quadratic equation?

1. x² + 2x + 1 = (4 - x)² + 3
2. -2x² = (5 - x)(2x - 2/5)
3. (k + 1)x² + (3/2)x = 7 where k = -1
4. x² + 3x = (-1) (1 - 3x)²

☛ Also Check: NCERT Solutions for Class 10 Maths Chapter 4

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NCERT Exemplar Class 10 Maths Exercise 4.1 Problem 1

Which of the following is a quadratic equation, a. x² + 2x + 1 = (4 - x)² + 3, b. -2x² = (5 - x)(2x - 2/5), c. (k + 1)x² + (3/2)x = 7 where k = -1, d. x³ - x² = (x - 1)³

Summary:

x³ - x² = (x - 1)³ is a quadratic equation

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