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Quadratic Equations

Long Answer Type

21. Find two consecutive positive integers, sum of whose square is 365.

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22. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm. Find the other two sides.

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Short Answer Type

23. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was Rs. 90. Find the number of articles produced and the cost of each article.

Let the number of articles produced be x
Therefore, cost of production of each article (in Rs.) on that particular day = Rs. (2x + 3)
So, total cost of production that day = Rs. x(2x + 3)
According to question,
x(2x+3) = 90
$rightwards double arrow$ $2 straight x squared plus 3 straight x equals 90$
$rightwards double arrow$ $space 2 straight x squared plus 3 straight x minus 90 equals 0$
$rightwards double arrow$ $space space space space space space space 2 straight x squared plus 15 straight x minus 12 straight x minus 90 space equals space 0$
$rightwards double arrow$ $straight x left parenthesis 2 straight x plus 15 right parenthesis minus 6 left parenthesis 2 straight x plus 15 right parenthesis space equals 0$
$rightwards double arrow$ $left parenthesis straight x minus 6 right parenthesis space left parenthesis 2 straight x plus 15 right parenthesis space equals space 0$
$rightwards double arrow$ $straight x minus 6 space equals space 0$
or $space space 2 straight x plus 15 space equals space 0$
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or $2 straight x space equals space minus 15$
$rightwards double arrow$ $straight x equals fraction numerator negative 15 over denominator 2 end fraction$
Since $straight x equals fraction numerator negative 15 over denominator 2 end fraction$ is not possible
So, x = 6
Hence, number of articles produced be x = 6 And cost of production of each article = Rs. (2x + 3)= Rs. (2 x 6 + 3) = Rs. (15).