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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.

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24. Find the roots of the following quadratic equations, if they exist by the method of completing the square:
2x² –7x + 3 = 0

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25. Find the roots of the following quadratic equations, if they exist by the method of completing the square:

2 straight x squared plus straight x minus 4 space equals space 0

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26. Find the roots of the following quadratic equations, if they exist by the method of completing the square:

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

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27. Find the roots of the following quadratic equations, if they exist by the method of completing the square:
2x²+x+ 4 = 0.

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Long Answer Type

28. Find the roots of the quadratic equations given in Q. 1, above by applying the quadratic formula.

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29. Find the roots of the factors of equations:
$straight x minus 1 over straight x equals 3 comma space space straight x not equal to 0.$

We have,
$straight x minus 1 over straight x equals 3$
$rightwards double arrow$ $fraction numerator straight x squared minus 1 over denominator straight x end fraction equals 3$
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$rightwards double arrow$ $straight x squared minus 3 straight x minus 1 space equals space 0$
Here, $straight a equals 1 comma space space straight b equals negative 3 space and space straight c space equals space minus 1$
Now, $straight b squared minus 4 ac equals left parenthesis negative 3 right parenthesis squared minus 4 left parenthesis 1 right parenthesis left parenthesis negative 1 right parenthesis space space space space space space space space space space space space space space space equals 9 plus 4 equals 13 Since comma space space straight b squared minus 4 ac greater than 0$
Therefore, quadratic equation x² – 3x – 1 =0 has distinct roots, i.e. α and β.
Now, $straight x equals fraction numerator negative straight b plus-or-minus square root of straight b squared minus 4 ac end root over denominator 2 straight a end fraction space space equals space fraction numerator negative left parenthesis negative 3 right parenthesis plus-or-minus square root of 13 over denominator 2 left parenthesis 1 right parenthesis end fraction equals fraction numerator 3 plus-or-minus square root of 13 over denominator 2 end fraction$

So, $space space space straight alpha space equals space fraction numerator 3 plus square root of 13 over denominator 2 end fraction$

and $<pre>uncaught exception: mkdir(): Permission denied (errno: 2) in /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php at line #56mkdir(): Permission denied in file: /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php line 56 #0 [internal function]: _hx_error_handler(2, 'mkdir(): Permis...', '/home/config_ad...', 56, Array) #1 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/util/sys/Store.class.php(56): mkdir('/home/config_ad...', 493) #2 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/FolderTreeStorageAndCache.class.php(110): com_wiris_util_sys_Store->mkdirs() #3 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/RenderImpl.class.php(231): com_wiris_plugin_impl_FolderTreeStorageAndCache->codeDigest('mml=<math xmlns...') #4 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/lib/com/wiris/plugin/impl/TextServiceImpl.class.php(59): com_wiris_plugin_impl_RenderImpl->computeDigest(NULL, Array) #5 /home/config_admin/public/felixventures.in/public/application/css/plugins/tiny_mce_wiris/integration/service.php(19): com_wiris_plugin_impl_TextServiceImpl->service('mathml2accessib...', Array) #6 {main}</pre>$
Therefore, the required roots are
 $fraction numerator 3 plus square root of 13 over denominator 2 end fraction space and space fraction numerator 3 minus square root of 13 over denominator 2 end fraction.$

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30. The sum of the reciprocals of Rehman’s age (in years) 3 years ago and 5 years from now is

space 1 third.

Find his present age.

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