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The answer is 52.

One of the five quadratics below has a repeated root.     

(The other four have distinct roots.) What is the repeated root?     

To have a repeated root, a and c both have to be positive squares.     

There is only one of the listed quadratics that fits this description.     

     x² – 28x + 196 factors to (x – 14)(x – 14)   ——>>   x  =  +14     

_.     

The answer is 5/4.

The solution is (7,1).

The equation x^2 + y^2 + 10x - 8y = 59 can be written in the     

form (x - h)^2 + (y - k)^2 = r^2, where h, k, and r are integers. Find the value of r^2.     

                    x² + y² + 10x – 8y  =  59     

Rearrange & add parentheses in     

preparation to complete squares.     

                    (x² + 10x    )  +  (y² – 8y    )  =  59     

To find c, if a=1, then c = (b/2)²     

                    (x² + 10x + 25)  +  (y² – 8y + 16)  =  59 + 25  + 16     

                    (x + 5)²  +  (y – 4)²  =  100    ——>>     r²  =  100     

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On the graph of y = (x + 2)^2 - 10, how many points are there whose coordinates are both negative integers?     

          y  =  (x + 2)² – 10     

          Considering only integers, it's obvious that when x < –6     

          then y goes positive, so with so few let's just try them all.     

          when x = –1,  y  =    –9     

          when x = –2,  y  =  –10     

          when x = –3,  y  =    –9     

          when x = –4,  y  =    –6     

          when x = –5,  y  =    –1        so there are five such points.     

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make x the subject of the formula y=a(x+b)     

                                             y  =  a(x + b)     

                                         y / a  =  x + b     

                                (y / a) –  b  =  x     

By convention, the subject is usually located     

on the left, although it doesn't have to be.     

                                so,         x  =  (y / a) –  b     

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There are 12 ordered pairs of integers (x,y) that satisfy x^2 + y^2 = 25.     

What is the greatest possible sum x+y if x = 5?      

                                                      x² + y²  =  25     

                              if x = 5             5² + y²  =  25     

                                                     25 + y²  =  25     

                                                             y²  =  25 – 25     

                                                             y²  =  0     

                                                             y    =  0     

                                                         x + y  =  5     

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Compute
1 + \frac{3}{6} + \frac{5}{6^2}     

                                                         1   +  ³/₆  +  ⁵/₃₆     

using a common denominator:      ³⁶/₃₆ + ¹⁸/₃₆ + ⁵/₃₆     

                                                       ⁵⁹/₃₆  =  1 ²³/₃₆     

_.     

the answer you are looking for is \(\frac{-505}{181}\)

What are all solutions to the equation x^2 + 3x - 10 = 18x - 15?     

                                                            x² + 3x – 10  =  18x – 15     

                                                            x² – 15x + 5  =  0     

Good luck factoring that . . .     

need the quadratic formula     

                                                                     – b  +  sqrt( b² – 4ac )     

                                                            x  =  –––––––––––––––––––     

                                                                                 2a     

                                                            x  =  [ 15 + sqrt( 225 – 20 ] / 2     

                                                            x  =  ¹/₂ • [ 15 + sqrt(205) ]     

                                                            x  =  14.65891         

                                                            x  =    0.34109     

_.     

The line is x = 4, and the equation of the circle is x^2 + y^2 = 25.     

                                         x² + y²  =  25     

                                         4² + y²  =  25     

                                         16 + y²  =  25     

                                                 y²  =  9     

                                                 y   =  +3     

                            AB  =  3 – (–3)  =  3 + 3     

                                               AB  =  6     

_.     

               60 + x             100 + x     

             ––––––––   =   ––––––––       

              100 + x            140 + x     

    (140 + x)(60 + x)  =  (100 + x)(100 + x)     

    x² + 200x + 8400  =  x² + 200x + 10000     

    The next step would be 8400 = 10000     

    so the problem has no solution.     

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The quantity inside the sqrt  must be ≥ 0

So

x^2 -8x + 5  ≥  0

Find the two roots of this quadratic

x  =  [ 8 +/- sqrt (64 -20] / 2

x = [ 8 + sqrt 44 ] /2       and   x  = [ 8 -sqrt 44] /2

Since x^2 -8x + 5  is a parabola that turns upward....the solution includes the roots (because of the ≥ sign) and also includes the intervals outside the roots  →

( -inf , (8 -sqrt 44)/2 ]  U  [ (8 + sqrt 44 ) / 2 , inf ) = the domain

An angle measures 178° less than the measure of its supplementary angle plus 56 degrees. What is the measure of each angle?     

Call "x" one of the supplemantary angles.     

Then, "180 – x" would be the other angle.     

      x + 178  =  (180 – x) + 56     

      x + 178  =  180 – x + 56     

           x + x  =  180 + 56 – 178     

               2x  =  58     

                 x  =  29     

     180 – 29  =  151     

     so the two angles are 29^o and 151^o     

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Adam has 16.45 kg of flour, and he uses 6.4kg to make hot cross buns .The remaining flour is exactly enough to make 15 batches of scones.How much flour,in kg will be in a batch of scones?     

          16.45     original amount of flour     

            6.40     used to make hot cross buns     

          10.05     remaining flour     

          10.05                         enough to make 15 batches of scones     

          10.05 / 15  =  0.67     what it takes to make one batch of scones     

_.     

A bag contains 10 counters. 6 of them are white. a counter is taken at random and not replaced. A second counter is taken out of the bag at random. Calculate the probability that both two counters are white.     

Probability first counter is white                6 / 10     

Probability second counter is white           5 / 9     

Probability both conditions occurring       (6 / 10) • (5 / 9)  =  30 / 90     

                                                                               reduces to 1 / 3    or  33 ¹/₃ %       

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Alex chooses a number at random from the set {1, 2, 3, 4, 5}.  Winnie also chooses a number at random from the same set.  (They can choose the same number.)  What is the probability that the product of their numbers is at least 4?     

The following are the 25 pairs that could occur.     

1 1    1 2    1 3    1 4    1 5     

2 1    2 2    2 3    2 4    2 5     

3 1    3 2    3 3    3 4    3 5     

4 1    4 2    4 3    4 4    4 5     

5 1    5 2    5 3    5 4    5 5     

The highlighted pairs are the 20 that multiply to at least four.     

The probability of choosing a highlighted pair is 20 / 25 which is 80%.     

_.     

The correct answer is 250.

The 450 juniors at East HS drink a total of 1500 cartons of milk per 5-day week. If the average senior drinks milk at the same rate as the average junior, what is the average number of total cartons of milk the 600 seniors drink each school day?     

1500 cartons for 5 days works out to 300 cartons for 1 day     

spread among 450 juniors, so each junior drinks ²/₃ of a carton per day.     

The seniors drink milk at the same rate; however, there are 600 of them,     

so the senior class drink ²/₃ x 600  =  400 cartons for 1 day     

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