Bosco

A terminal zero is a $0$ that appears at the end of a number. For example, the number $3,800$ has two terminal zeros.    
How many terminal zeroes does $40 \cdot 6 \cdot 75 \cdot 12$ have?    

40 • 6 • 75 • 12 = 216,000     Three    

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What is the smallest positive integer that has exactly $2$ perfect square divisors?    

That would be 4 ... namely, 1² and 2².    

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Fill in the blanks with numbers to make a true equation.    
3x^2 + 12x - 4 - 2x^2 + 6x + 7 = ___ (x + ___ )^2 + ___    

Combine like terms on the left          x² + 18x + 3  =        (x +       )² +          

Complete the square on the left     ( x² + 18x + 81) – 78  =    1  (x +  9  )² +  –78     

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Since the roots are 1 and – 1    

then the factors are (x – 1) and (x + 1)     

Multiply (x – 1) times (x + 1) to obtain x² – 1    

Therefore, fill the blank spaces with 0 and – 1    

x² + (0) x + (– 1)    

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                                      a + ab =   250    

                                      a – ab = –240    

just add them               2a         =     10    

                                             a  =  5    

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

When a quadratic is arranged in standard format, ax² + bx + c = 0,    

the discriminant is given by b² – 4ac.    

Combine like terms in the quadratic of interest and arrange in standard format.    

3x² + 4x + 10   

b² – 4ac    

4² – (4)(3)(10)    

16 – 120    

–104 is the discriminant    

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Find the largest integer k such that the equation 5x^2 - kx + 8 - 2x^2 + 25 =0 has no real solutions  

Combine like terms and arrange equation    

in standard format ax² + bx + c = 0                3x² + (–k) x + 33 = 0    

The discriminant is defined as b² – 4ac          (–k)² – (4)(3)(33)    

                                                                          k² – 396    

An equation has no real solutions when the discriminant is negative.    

This is because in the quadratic formula, you calculate the square root of the discriminant.    

Therefore k² must be smaller than 396 to result in a negative discriminant.    

sqrt(396) is 19.9,  The next smaller integer is 19.  So, k = 19 answers the prolblem.         

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

Given                               3x² + 17x + 15  =  2x² + 21x + 12 – 5x² + 17x + 34    

Combine like terms          6x² . . .   We can stop here; no need to bother with the other terms.    

The highest order term is the "squared term" and it is positive, which means that the curve    

is a parabola opening upward, for which there is no largest value of x.  It continues forever.    

We could say it approaches infinity, with the caveat that infinity can never be used as a value.    

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

I answered this once before and my answer was good,

but then I saw a better answer.  Here's the better answer.    

1/2  +  1/3  +  1/4    

6/12 + 4/12 + 3/12  =  13/12    

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