Trying to find the right answer to this..

R.H.S. = (x – 2) 3 = a 3 – 2 3 – 3x.2 (a – 2) = x 3 – 8 – 6x 2 + 12x = a 3 – 6x 2 + 12x – 8 ∴ x 3 – 4x 2 – x + 1 = x 3 – 6x 2 + 12x – 8 ⇒ –4x 2 + 6x 2 – x – 12x + 1 + 8 = 0 ⇒ 2x 2 – 13x + 9 = 0 ∵ It is of the form ax 2 + bx + c = 0 where a = 2, b = –13, c = 9. Hence, the given equation is 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)

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

R.H.S. = (x – 2)³= a³ – 2³ – 3x.2 (a – 2)
= x³ – 8 – 6x² + 12x
= a³ – 6x² + 12x – 8
∴ x³ – 4x² – x + 1 = x³ – 6x² + 12x – 8
⇒ –4x² + 6x²– x – 12x + 1 + 8 = 0
⇒ 2x²– 13x + 9 = 0
∵ It is of the form ax² + bx + c = 0 where a = 2, b = –13, c = 9.
Hence, the given equation is a quadratic equation.

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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.

220 Views