Bosco

Find the largest value of $x$ such that $3x^2 + 17x + 15 = 2x^2 + 21x + 12 - 5x^2 + 17x + 34.$   

rearrange terms for clarity              (3x² – 2x² + 5x²) + (17x – 21x – 17x) + (15 – 12 – 34)  =  0   

combine like terms                          6x² – 21x – 31  =  0   

A parabola that opens upward has no largest value.   

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5 • 6 • 7 • 8 = 1680.  Then add 1 to it.      

Brandon picked 1681 apples   

check answer   

                          1681 / 5  =  335 R 1   

                          1681 / 6  =  280 R 1   

                          1681 / 7  =  240 R 1   

                          1681 / 8  =  210 R 1   

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The numbers $24^2 = 576$ and $56^2 = 3136$ are examples of perfect squares that have a units digits of $6.$   
If the units digit of a perfect square is $5,$ then what are the possible values of the tens digit?   

If a number ends with 5, its square will end with 25.  

So the only possible value of the tens digit is 2.   

How to square a number that ends with a 5.   

Separate the 5 out of the number.  

Change that 5 to 25.   

Add 1 to the number that is left. 

Multiply that by the original number.   

Stick the 25 to the end of the product.   

Examples:  

   15²                              45²                              75²                              125²       

   1                  5              4                  5             7                  5              12                  5   

   1                25              4                25             7                25              12                25    

   1 x (1+1)    25              4 x (4+1)    25             7 x (7+1)    25              12 x (12+1)  25    

   1 x 2           25              4 x 5          25             7 x 8           25              12 x 13        25     

   225                              2025                           5625                             15625          

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In the example, the divisors of 100 include 1 and 100.  

Using that standard, the smallest integer would be 4    

because its perfect square divisors are 1 and 4.  

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               1    

            –––––  =  0.001023541453429   

              977   

            That's as far as my calculator reads out.   

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Brandon spends his afternoon picking apples from an orchard.  He notices that if he groups the apples he picked by fives, six, or sevens, then he ends up with $1$ left over.  If he groups by eights, he has $1$ left over.  If Brandon knows that he picked between $1000$ and $2000$ apples, how many did he pick?   

I multiplied 5 • 6 • 7 • 8 and got 1680.  Then I added 1 to it.   

                                              Brandon picked 1681 apples   

check answer   

                                              1681 / 5  =  335 R 1   

                                              1681 / 6  =  280 R 1   

                                              1681 / 7  =  240 R 1   

                                              1681 / 8  =  210 R 1   

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Find the sum of all values of $k$ for which $x^2 + kx - 9x + 25 - 9$ is the square of a binomial.   

                                                                       x² + (kx – 9x) + 16   

factor out x                                                      x² + (k–9)(x) + 16

When posed as ax² + bx + c, for there to be     

a double root, also called a repeated root,   

the term c must be a square and b = 2sqrt(c)  

Square root of 16 = +4 so b = (2)(+4) = +8                  

when +4 is positive                                         k – 9 = +8        

                                                                       k = +8 + 9   

                                                                       k = 17  

when +4 is negative                                        k – 9 = –8   

                                                                       k = –8 + 9   

                                                                       k = 1   

the sum of all values of k                                sum = 17 + 1   

                                                                       sum = 18   

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There is no graph shown.  Relying only on the three points provided, one   

parabola that passes through the points (–2,8), (0,0), and (2,8) is y = 2x²    

If that is correct, then a + b + c  =  2 + 0 + 0  =  2.   

Those three points, and the vertex at (0,0), indicate that the parabola    

is symmetrical to the y-axis and that the equation has a double root.  

_.

Yes, it is correct.  I was still typing when your answer was   

posted, but we both solved it the same way.  Good work.   

_,

Find the constant k such that the quadratic 2x^2 + 3x + 8x - x^2 + 4x + k has a double root.  

                                                                    2x² + 3x + 8x – x² + 4x + k   

combine like terms                                         x² + 15x + k   

for a quadratic to have a double root,   

which is also called a repeated root,   

its discriminant must equal zero

when a quadratic is in the form ax² + bx +c   

its discriminant is the quantity b² – 4ac            b² – 4ac   

                                                                       15² – (4)(1)(k)   

                                                                       225 – 4k    

solve for k                                                          k  =  (225 / 4)    

                                                                           k  =  56.25    

by the way, the quadratic in this problem factors to (x + 7.5)(x + 7.5)   

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