Bosco

Find the sum of the squares of the roots of $2x^2+4x-1=x^2-8x+3$.  

                                           2x² + 4x – 1  =  x² – 8x + 3  

Combine like terms               x² + 12x – 4  =  0  

Complete the square            x² + 12x + 36  =  40  

                                             (x + 6)²  =  40  

                                              x + 6  =  +sqrt(40)  

                                                    x  =  +sqrt(40) – 6  

Square each root      [ +sqrt(40) – 6 ]²  =  40 – 12sqrt(40) + 36      (eq 1)   

                                  [ –sqrt(40) – 6 ]²  =  40 + 12sqrt(40) + 36      (eq 2)   

Add (eq 1) & (eq 2)                                    80                      + 72   

                              Sum of squares  =  152    

_.

For how many three-digit positive integers is the sum of the digits equal to 5?  

104      203      302      401      500  

113      212      311      410  

122      221      320  

131      230  

140  

If I didn't miss any, I get 15 of them.    

_.

Laverne starts counting out loud by ${}5$'s.  She starts with $2$.  As Laverne counts, Shirley sums the numbers Laverne says.  When the sum finally exceeds $20,$ Shirley runs screaming from the room.  What number does Laverne say that sends Shirley screaming and running?  

Laverne     Shirley   

       2              2  

       7              9   

     12            21  

_.

Hi Chris, 

So, I blundered into the right answer but for the wrong reason. 

All the wrong reasons.  Thanks for posting the correct answer. 

Ron  

_.

In a rhombus, all the side lengths are $12,$ and one of the angles is equal to $20^\circ.$ Find the area of the rhombus.   

Visualize, or preferably draw, the rhombus, leaning sharply to the right.  

Drop a perpendicular from the upper left vertex to the bottom side. 

This has made a right triangle with a 20^o angle and the hypotenuse 12.  

Use trig to find that height..       sin(20^o)  =  height / 12  

                                                    0.3420  =  height / 12 

                                                      height  =  0.3420 • 12  =  4.104

                                            Area_rhombus  =  base • height 

                                                        Area  =  12 • 4.104 

                                                        Area  =  49.25  

Note that I rounded the decimal values that I wrote down.    

If you calculate it using those values, you'll get 49.248.   

I calculated it without rounding, that's where I got 49.25.  

_.

My square patio is tiled with square tiles, all the same size.  All the tiles are gray, except the tiles along the two diagonals, which are all yellow.  (The corners are yellow, the center is yellow, and all the tiles along the diagonal in between are yellow.)  If there are $20$ yellow tiles, how many gray tiles are there?  

I can make it work for 21 yellow tiles, but not 20.  

   Y   G   G   G   G   G   G   G   G   G   Y  

   G   Y   G   G   G   G   G   G   G   Y   G 

   G   G   Y   G   G   G   G   G   Y   G   G 

   G   G   G   Y   G   G   G   Y   G   G   G 

   G   G   G   G   Y   G   Y   G   G   G   G 

   G   G   G   G   G   Y   G   G   G   G   G 

   G   G   G   G   Y   G   Y   G   G   G   G 

   G   G   G   Y   G   G   G   Y   G   G   G 

   G   G   Y   G   G   G   G   G   Y   G   G 

   G   Y   G   G   G   G   G   G   G   Y   G 

   Y   G   G   G   G   G   G   G   G   G   Y  

_.

Let $ABCD$ be a square with side length $1.$ A laser is located at vertex $A,$ which fires a laser beam at point $X$ on side $\overline{BC},$ such that $BX = \frac{1}{2}.$  The beam reflects off the sides of the square, until it ends up at another vertex; at this point, the beam will stop.  Find the length of the total path of the laser beam.  

The beam starts at corner A and bounces off the middle of side BC.   

The next place the reflected beam should strike would be corner D. 

The beam makes a right triangle ABX with AX being the hypotenuse. 

Using Pythagoras Theorem,                                   AX²  =  1² + 0.5²  

                                                                               AX²  =  1.5  

                                                                                 AX  =  1.118   

The reflected beam would be the same length.  

So, totalling AX + XD,

the total length of the beam is                  1.118 + 1.118  =  2.236    

_.

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.    

The length of the median is equal to the average of the two bases.  

So, with one base 12, and the other base 16 longer, we have  

                  12 + (12 + 16)        40  

median  =  ––––––––––––  =  ––––  =  20  

                             2                   2  

The short base has to be 12, because 12 can't be 16 greater than another positive number. 

_.