Bosco

Kaspar is buying apples and oranges from the market. The total cost of 4 apples and 3 oranges is $11, while the total cost of 2 apples and 5 oranges is $13. What is the cost of each apple and each orange?  

                                                  4A + 3G  =  11     (eq 1)  

                                                  2A + 5G  =  13     (eq 2) 

Multiply (eq 2) by 2                    4A + 10G  =  26   (eq 3)  

Subtract (eq 1) from (eq 3)                  7G  =  15  

                                                              G  =  15 / 7  =  2.14   (one orange)  

Plug the value of an orange  

back into (eq 1)                          4A + (3)(2.14)  =  11  

                                                                     4A  =  11 – 6.42  =  4.58  

                                                                       A  = 4.58 / 4  =  1.14   (one apple)  

Check answers  

(eq 1)                            4 • (1.14) + 3 • (2.14)  =  10.98   close enough to 11  

(eq 2)                            2 • (1.14) + 5 • (2.14)  =  12.98   close enough to 13  

The answers check out, considering that the calculations required rounding.  

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                                                x² – 5x + p + 2x² – 8x  

                                                Combine like terms, arrange in standard form ax² + bx + c  

                                                3x² – 13x + p  

                                                 An equation is a square (another way to say this is that  

                                                 it has a double root) when its discriminant equals zero   

                                                 b² – 4ac =  0  

                                                 (–13)² – (4)(3) • p  =  0  

                                                 169 – 12p  =  0  

                                                        – 12p  =  – 169  

                                                               p  =  169 / 12  

Checked that the answer is a square by using Desmos to  

plot the curve of y = 3x² – 13x + 169/12 and confirming   

that it crosses touches the x-axis at only one point. 

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

                                                2x² + 3x + 8k + 3x² + 2x + k    

                                                combine like terms, and arrange in standard form ax² + bx + c  

                                                 5x² + 5x + 9k  

                                                 an equation has a double root when its discriminant equals zero   

                                                 b² – 4ac =  0  

                                                 5² – (4)(5) • 9k  =  0  

                                                         25 – 180k  =  0  

                                                              – 180k  =  – 25  

                                                                       k  =  25 / 180  

                                                                       k  =  5 / 36  

_.

                                      (8/125)*___ = 5/2  

                                         8          x          5  

                                      ––––  •  –––  =  –––  

                                       125        1          2  

divide both sides by 8 / 125  

                                                             5           8  

                                                    x  =  –––  ÷  ––––   

                                                             2         125  

to divide by a fraction invert and multiply  

                                                             5         125  

                                                    x  =  –––  •  ––––   

                                                             2           8    

                                                               5         125          625  

                                                    x   =   –––  •  ––––   =   –––––   

                                                               2           8             16  

.

Find the discriminant of the quadratic $5x^2 - 2x + 8 - 3x^2 + 8x + 12.$  

                                                              5x² – 2x + 8 – 3x² + 8x + 12   

Combine like terms                                2x² + 6x + 20  

When in the form ax² + bx + c the discriminant is b² – 4ac  

                                                               6² – (4)(2)(20)  

                                                               36 – 160  

                                                               –124 is the discriminant  

_.

(11 - 7i)*(2 + i)  

22  +  11i  –  14i  –  7i²      (remember that i² = –1)  

22  +  11i  –  14i  +  7  

29 – 3i  

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Seems pretty obvious that when k = 3 it cancels out the –3i and you end up with just 15.  

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97² = 9409   (c)  

90² = 8100   (a or b)  

17² =   289   (b or a)  

a² + b² must = c²  but, as you can see from the above, it does not.  

The Pythagorean triple with 97 as the hypotenuse is not (90,17,97).  

I recommend always checking an answer, if possible, before posting.  

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