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Quadratic Equations

Short Answer Type

41. Find the values of k for each of the following quadratic equations, so that they have equal roots.

space 2 straight x squared plus kx plus 3 equals 0

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42. Find the values of k for each of the following quadratic equations, so that they have equal two roots.
kx (x – 2) + 6 = 0

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Long Answer Type

43. Is it possible to design a rectangular Mango grove whose length is twice its breadth and the area is 800 m²? If so, find its length and breadth.

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44.

Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.

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45.

Is it possible to design a rectangular park of perimeter 80 m and area 400 m²? If so, find its length and breadth.

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Short Answer Type

46. State whether the following is quadratic equation:
(x – 2)² (x – 1) = (x – 4) (x + 3)

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47. State whether the following is quadratic equation:
(x – 4) (x – 3) = (x – 3) (x² – 4)

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48. State whether the following is quadratic equation:

square root of 2 straight x squared plus 4 straight x plus 5 square root of 2 plus 3 equals 0

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49. Write the nature of the roots of the following quadratic equation:

space space straight x squared minus 1 half straight x minus 4 equals 0.

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50. Write the nature of the roots of the following quadratic equation:
$3 straight x squared minus 2 straight x plus 1 third equals 0$

3 straight x squared minus straight x plus 1 third equals 0

Here,

straight a equals 3 comma space space straight b space equals space minus 2 space space and space straight c space equals space 1 third

straight D equals straight b squared minus 4 ac

equals left parenthesis negative 2 right parenthesis squared minus 4 left parenthesis 3 right parenthesis open parentheses 1 third close parentheses

equals 4 minus 4 space equals space 0

∵ D = 0
Hence, roots are real and equal.

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