Bosco

The quadratic equation x² – 5x + t = 3x has only positive integer roots. Find the average of all distinct possible values of t.  

Subtract 3x from both sides        x² – 8x + t = 0    

Note that the resulting   

equation will factor.                     (x – 1)(x – 7) = 0   gives us t =   7  

                                                   (x – 2)(x – 6) = 0   gives us t = 12  

                                                   (x – 3)(x – 5) = 0   gives us t = 15  

                                                   (x – 4)(x – 4) = 0   gives us t = 16  

                                            7 + 12 + 15 + 16          50

Average of possible t's  =  –––––––––––––––  =  ––––  =  12.5  

                                                       4                        4  

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The real numbers x and y satisfy x (x+y) = 27 and (x+y) = 54. What is the value of (x+y)^2?  

If  (x+y) = 54  then  (x+y)² = 54²  =  2916   

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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's         Shirley's  

Counting            Sums  

        2                    2  

        7                    9  

      12                  21      I don't get it.  The running and screaming part, I mean. 

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8 squared is 64 so every integer whose absolute value is less than 8 satisfies the problem. 

These are –7, –6, –5, –4, –3, –2, –1, 0, 1, 2, 3, 4, 5, 6, and 7  

I count 15 of them.  

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If $t$ is a real number, what is the maximum possible value of the expression $-t^2 + 8t -4 +5t^2 - 4t + 18$?  

When you combine like terms, you get 4t² plus some other   

stuff, but the square is all we're interested in for this problem. 

The 4t² indicates that this curve is a parabola that opens  .  

upward.  Since it opens upward, there is no maximum value,   

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What is the solution to x+y=0?   

          x + y = 0   

               x  =  –y   

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angle PCD = 75^o  

Grab a scrap of paper and sketch it. 

The line from P to C is the base of a isoceles triangle. 

Its equal sides are a side of the square and a side of the triangle.  

The big angle of the isoceles triangle is the sum of the 60^o of the  

equilateral triangle and the 90^o of the square, for a total of 150^o.   

The sum of the interior angles of any triangle is 180^o and so that   

leaves 30^o for the two small angles to share, namely 15^o each.  

At corner C, subtract that 15^o from the 90^o of the square, leaving 75^o.  

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