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L.H.S. = (x + 2) 3 = x 3 + 3x 2 .2 + 3x.(2) 2 +8 = x 3 + 6x 2 + 12x + 8 R.H.S. = 2x(x 2 – 1) = 2x 3 – 2x ∴ x 3 + 6x 2 + 12x + 8 = 2x 3 - 2x ⇒ x 3 – 2x 3 + 6x 2 + 12x + 2x + 8 = 0 ⇒ –x 3 + 6x 2 + 14x + 8 = 0 ∵ It is not of the form ax 2 + bx + c = 0. Hence, the given equation is not a quadratic equation.

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Quadratic Equations

Short Answer Type

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

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

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

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4. Check whether the following are quadratic equations:
(x – 3) (2x + 1) = x(x + 5)

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5. Check whether the following are quadratic equations:
(2x –1) (x–3) = (x + 5)(x–1)

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

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

L.H.S. = (x + 2)³= x³ + 3x².2 + 3x.(2)² +8
= x³ + 6x² + 12x + 8
R.H.S. = 2x(x² – 1) = 2x³– 2x
∴ x³ + 6x² + 12x + 8 = 2x³ - 2x
⇒ x³ – 2x³ + 6x² + 12x + 2x + 8 = 0
⇒ –x³ + 6x² + 14x + 8 = 0
∵ It is not of the form ax² + bx + c = 0.
Hence, the given equation is not a quadratic equation.

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8. Check whether the following are quadratic equations:
x³ – 4x² – x + 1 = (x – 2)³.

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Long Answer Type

9. Represent the following problem situations in the form of quadratic equations:
The area of a rectangular plot is 528 m². The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.

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Short Answer Type

10. Represent the following problem situations in the form of quadratic equations:
The product of two consecutive positive integers is 306. We need to find the integers.

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