Bosco

Find the discriminant of the quadratic 5x^2 - 2x + 8 - 2x^2 + 6x + 2 - 2x^2 + 4x - 1.    

First, collect and combine like terms   

and arrange them as ax² + bx + c           (5x² – 2x² – 2x²) + (–2x + 6x + 4x) + (+8 + 2 – 1)    

                                                                  x² + 8x + 9    

The discriminant is b² – 4ac                     D  =  8² – (4)(1)(9)    

                                                                 D  =  64 – 36    

                                                                 D  =  28    

_.   

Fill in the blanks with three distinct positive integers.    
1/___ + 1/___ + 1/___ = \frac{13}{12}    

                     1          1          1           39         13     

                  –––– + –––– + ––––  =  ––––  =  ––––    

                     1         18        36          36         12     

_.   

Let $IJKLMN$ be a hexagon with side lengths $IJ = LM = 3,$ $JK = MN = 3,$ and $KL = NI = 3$. Also, all the interior angles of the hexagon are equal. Find the area of hexagon $IJKLMN$.   

                                                                                         3 • sqrt(3)    

Area of a regular hexagon in terms of its sides is     A  =  –––––––– • a²       where a is the length of a side

                                                                                                2    

                                              A  =  (3)² • [ 3 • sqrt(3) ] / (2)         

                                              A  =  27 • sqrt(3) / 2  =  23.38        

The interior angles of a polygon form an arithmetic sequence. The difference between the largest angle and smallest angle is $56^\circ$. If the polygon has $3$ sides, then find the smallest angle, in degrees.    

Call the three angles a, b, and c, in order of size.         

The problem states c – a = 56         ==>>   c = a + 56    

The angles are in an arithmetic

sequence, so b – a = (1/2)(56)         ==>>   b = a + 28    

The sum of the three angles of a    

triangle is 180, so a + b + c = 180    ==>>  (a) + (a + 28) + (a + 56)  =  180    

                                                                                             3a + 84  =  180    

                                                                                                     3a  =  96    

                                                                                                       a  =  32    

check answer    

                           a + b + c = 180   

32 + (32 + 28) + (32 + 56) = 180    

                                   180 = 180      ==>> True   

_.    

The product of two positive integers minus their sum is 39. The integers are relatively prime, and each is less than 2. What is the sum of the two integers?       

A pair that works is (11, 5) so 16 is the sum.   

Assuming that "less than 2" is a typo that should have been "less than 20".     

Another pair that works is (6, 9) but they aren't relatively prime.    

I found them by plotting xy – x – y = 39 on Desmos, then    

finding the points where both x and y are positive integers.      

_.   

The area of a trapezoid is $80$. The length of one base is $16$ units greater than the other base, and one base of the trapezoid is $12$. Find the length of the median of the trapezoid.   

        ( base_top + base_bottom )    

M  =  –––––––––––––––––––        

                        2    

M  =  [ 12 + (12+16) ] / 2  =  40 / 2  =  20    

The median is the average of the two bases.    

The area is irrelevant to solving this problem.    

_.   

edited to correct misspelled "irrelevant" 

         ( base_top + base_bottom )    

A  =  ––––––––––––––––––– • height    

                        2    

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