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Question
Which of the following is a quadratic equation?
Options
`x^2 + 2x + 1 = (4 - x)^2 + 3`
`-2x^2 = (5 - x)(x - 2/5)`
`(k + 1)x^2 + 3/2x` = 7, where k = –1
`x^3 - x^2 = (x - 1)^3`
Solution
x3 – x2 = (x – 1)3
Explanation:
The standard form of a quadratic equation is given by,
ax2 + bx + c = 0, a ≠ 0
(A) Given, x2 + 2x + 1 = (4 – x)2 + 3
x2 + 2x + 1 = 16 – 8x + x2 + 3
10x – 18 = 0
which is not a quadratic equation.
(B) Given, –2x2 = `(5 - x)(2x - 2/5)`
–2x2 = `10x - 2x^2 - 2 + (2x)/5`
50x + 2x – 10 = 0
52x – 10 = 0
which is not a quadratic equation.
(C) Given, `(k + 1)x^2 + 3/2x` = 7, where k = –1
`(-1 + 1)x^2 + 3/2x` = 7
3x – 14 = 0
which is not a quadratic equation.
(D) Given, x3 – x2 = (x – 1)3
x3 – x2 = x3 – 3x2(1) + 3x(1)2 – (1)3 ...[ ∵ (a – b)3 = a3 – b3 + 3ab2 – 3a2b]
x3 – x2 = x3 – 3x2 + 3x – 1
–x2 + 3x2 – 3x + 1 = 0
2x2 – 3x + 1 = 0
which represents a quadratic equation.
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