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L.H.S. = (2x – 1) (x – 3) = 2x 2 – 6x– + 3 = 2x 2 – 7x + 3 R.H.S. = (x + 5) (a – 1) = x 2 + 5x – x – 5 = x 2 + 4x – 5 ∴2x 2 – 7x + 3 = x 2 + 4x – 5 ⇒ 2x 2 –x 2 -7x–4x+ 3 + 5 = 0 ⇒ x 2 – 11x + 8 = 0 ∵ It is of the form ax 2 + bx + c = 0 where a = 1, b = –11, c = 8. 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)

533 Views

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)

L.H.S. = (2x – 1) (x – 3)
= 2x² – 6x– + 3
= 2x² – 7x + 3
R.H.S. = (x + 5) (a – 1)
= x² + 5x – x – 5
= x² + 4x – 5
∴2x² – 7x + 3 = x² + 4x – 5
⇒ 2x² –x²-7x–4x+ 3 + 5 = 0
⇒ x² – 11x + 8 = 0
∵ It is of the form ax² + bx + c = 0 where a = 1, b = –11, c = 8.
Hence, the given equation is a quadratic equation.

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

186 Views

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

156 Views

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

125 Views

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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