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Quadratic Equations

Long Answer Type

11. Represent the following problem situations in the form of quadratic equations:
Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.

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12.
A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Let the speed of the train = x km/hr Distance = 480 km
Time taken by the train to cover the distance of 480 km

equals space 480 over straight x hrs

Now, according to the given condition,

fraction numerator 480 over denominator straight x minus 8 end fraction minus 480 over straight x equals 3

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480 open square brackets fraction numerator 1 over denominator straight x minus 8 end fraction minus 1 over straight x close square brackets equals 3

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open square brackets fraction numerator straight x minus straight x plus 8 over denominator left parenthesis straight x minus 8 right parenthesis. straight x end fraction close square brackets equals 3 over 480

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straight x squared minus 8 straight x minus 1280 equals 0

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straight x squared minus 40 straight x plus 32 straight x minus 1280 equals 0

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left parenthesis straight x minus 40 right parenthesis left parenthesis straight x plus 32 right parenthesis space equals space 0

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straight x minus 40 space equals space 0 space space space or space space straight x space plus space 32 space equals space 0

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straight x equals space 40 space space space or space space space straight x space equals space 32.

But speed of the train cannot be –ve. x = 40. Hence, the speed of the train = 40 km/hr.

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Short Answer Type

13. Find the roots of the following quadratic equations by factorisation method:
x² – 3x – 10 = 0

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14. Find the roots of the following quadratic equations by factorisation method:
2x² + x – 6 = 0

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15. Find the roots of the following quadratic equations by factorisation method:

square root of 2 straight x squared plus 7 straight x plus 5 square root of 2 equals 0

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16. Find the roots of the following quadratic equations by factorisation method:

2 straight x squared minus straight x plus 1 over 8 equals 0

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17. Find the roots of the following quadratic equations by factorisation method:

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Long Answer Type

18. John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.

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19. A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of articles produced in a day. On a particular day, the total cost of production was Rs. 750. We would like to find out the number of toys produced on that day.

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