Check whether the following are quadratic equations:
(i) (x + 1)2 = 2(x - 3)
ii) x2 - 2x = (-2)(3 - x)
iii) (x - 2)(x + 1) = (x - 1)(x + 3)
iv) (x - 3)(2x +1) = x(x + 5)
v) (2x -1)(x - 3) = (x + 5)(x -1)
vi) x2 + 3x +1 = (x - 2)2
vii) (x + 2)3 = 2x (x2 -1)
viii) x3 - 4x2 - x +1 = (x - 2)3
Solution:
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.
(i) (x + 1)2 = 2(x - 3)
Since (a + b)2 = a2 + b2 + 2ab
x2 + 2x + 1 = (2x - 6)
x2 + 2x + 1 - (2x - 6) = 0
x2 + 2x + 1 - 2x + 6 = 0
x2 + 7 = 0
Here, the degree of x2 + 7 = 0 is 2.
∴ It is a quadratic equation.
(ii) x2 - 2x = (-2) (3 - x)
x2 - 2x = - 6 + 2x
x2 - 2x - 2x + 6 = 0
x2 - 4x + 6 = 0
Degree = 2
∴ It is a quadratic equation.
iii) (x - 2)( x + 1) = (x -1)( x + 3)
x2 - 2x + x - 2 = x2 + 3x - x - 3
x2 - x - 2 = x2 + 2x - 3
x2 - x - 2 - x2 - 2x + 3 = 0
-3x + 1 = 0
Degree = 1
∴ It is not a quadratic equation.
iv) (x - 3)(2x +1) = x ( x + 5)
2x2 + x - 6x - 3 = x2 + 5x
2x2 - 5x - 3 = x2 + 5x
2x2 - 5x - 3 - x2 - 5x = 0
x2 - 10x - 3 = 0
Degree = 2
∴ It is a quadratic equation.
v) (2x -1)(x - 3) = (x + 5)(x -1)
2x2 - 6x - x + 3 = x2 - x + 5x - 5
2x2 - 7x + 3 = x2 + 4x - 5
2x2 - 7x + 3 - x2 - 4x + 5 = 0
x2 - 11x + 8 = 0
Degree = 2
∴ It is a quadratic equation
vi) x2 + 3x +1 = (x - 2)2
x2 + 3x +1 = x2 - 4x + 4 [∵ (a - b)2 = a2 - 2ab + b2]
x2 + 3x + 1 - x2 + 4x - 4 = 0
7x - 3 = 0
Degree = 1
∴ It is not a quadratic equation.
vii) (x + 2)3 = 2x (x2 -1)
We know that, (a + b)3 = a3 + 3a2b + 3ab2 + b3
x3 + 3x2 (2) + 3(x)(2)2 + (2)3 = 2x3 - 2x
x3 + 6x2 + 12x + 8 = 2x3 - 2x
- x3 + 6x2 + 14x + 8 = 0
Degree = 3
∴ It is not a quadratic equation.
viii) x3 - 4x2 - x + 1 = (x - 2)3
x3 - 4x2 - x + 1 = x3 - 3(x)2(2) + 3(x)(2)2 - (2)3 [∵ (a - b)3 = a3 - 3a2b + 3ab2 - b3]
x3 - 4x2 - x + 1 = x3 - 6x2 + 12x - 8
x3 - 4x2 - x + 1 - x3 + 6x2 - 12x + 8 = 0
2x2 - 13x + 9 = 0
Degree = 2
∴ It is a quadratic equation
☛ Check: NCERT Solutions for Class 10 Maths Chapter 4
Video Solution:
Check whether the following are quadratic equations: (i) (x + 1)² = 2(x - 3) (ii) x² - 2x = (-2)(3 - x) (iii) (x - 2)(x + 1) = (x - 1)(x + 3) (iv) (x - 3)(2x +1) = x(x + 5) (v) (2x -1)(x - 3) = (x + 5)(x -1) (vi) x² + 3x +1 = (x - 2)² vii) (x + 2)³ = 2x (x² -1) viii) x³ - 4x² - x + 1 = (x - 2)³
NCERT Solutions Class 10 Maths Chapter 4 Exercise 4.1 Question 1
Summary:
The equations i) (x + 1)² = 2(x - 3) (ii) x² - 2x = (-2)(3 - x), ii) x² - 2x = (-2)(3 - x), iv) (x - 3)(2x +1) = x(x + 5), v) (2x -1)(x - 3) = (x + 5)(x -1) and viii) x³ - 4x² - x + 1 = (x - 2)³ are quadratic equations while iii) (x - 2)(x + 1) = (x - 1)(x + 3), vi) x² + 3x +1 = (x - 2)² and vii) (x + 2)³ = 2x (x² -1) are not quadratic equations.
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