Bosco

Let u and v be the solutions to 3x^2 + 5x + 7 = 2x^2 + 5x - 19.  Find \frac{u}{v} + \frac{v}{u}. 

                                                 3x² + 5x + 7  =  2x² + 5x – 19  

Combine like terms                     x²  =  –26  

therefore                                     x  =  + sqrt(–26)

                                                   Call u  =  + sqrt(–26)  

                                                   Call v  =  – sqrt(–26) 

 u       + sqrt(–26)                               v       – sqrt(–26)  

––  =  ––––––––––  =  –1                 ––  =  ––––––––––  =  –1  

 v        – sqrt(–26)                              u        + sqrt(–26)  

therefore                                       u/v + v/u  =  –1 + (–1)  

                                                      u/v + v/u  =  –2   

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Since the quadratic's roots are +5 and –5 then the factors are (x – 5)(x + 5).  

That makes the product x² – 25.  Therefore the blanks are filled by 0 and –25.  

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Let a and b be the solutions to 5x^2 - 11x + 4 = 2x^2 - 7x + 9 + 7x^2 - 14x - 11. Find 1/a + 1/b.  

                                                         5x² – 11x + 4  =  2x² – 7x + 9 + 7x² – 14x – 11   

Combine like terms                           5x² – 11x + 4  =  9x² – 21x – 2   

Move everything to the left side      –4x² + 10x + 6  =  0   

Divide everything by –2                     2x² – 5x – 3  =  0  

Factor the quadratic                          (2x + 1)(x – 3)  =  0    

                                                              x = –1/2  and  x = 3      These are the values of a & b.  

                                                                1           1  

                                                             ––––  +  ––––  

                                                             –1/2          3    

                                                              –2           1  

                                                             ––––  +  ––––  

                                                                1            3  

                                                              –6           1                  5  

                                                             ––––  +  ––––   =    – ––––  

                                                                3            3                  3   

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I'm thinking of a number. My number has the property that its square is equal to $270$ more than five times the number. Enter all numbers that have this property.  

                                                   x²  =  5x + 270  

Get everything on the left           x² – 5x – 270  =  0  

I couldn't figure out how to factor this, so I used Desmos to plot y = x² – 5x – 270   

which confirmed that the roots are not integers.  Where the parabola crosses the  

x-axis shows the roots are a little smaller than –14.1 and a little larger than +19.1.  

You will have to use the quadratic formula if you want exact numbers, if you can   

use exact to describe a number that includes plus/minus the square root of 1105.  

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Find all values of x such that x² – 5x + 4 = 0.  If you find more than one value, then list your solutions, separated by commas.  

                                                   x² – 5x + 4 = 0  

This is a quadratic equation,  

and an easy one, too.  

It factors to                                   (x – 1)(x – 4) = 0   

Therefore, the values of x are       x = 1  &  x = 4  

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Draw a line from the center to the point of tangency.  

This is one leg of a right triangle, its length is 8.  

Draw a line from the center to one end of the chord.  

This is that triangle's hypotenuse, its length is 17.  

By Pythagoras' Theorem the other leg of the triangle, 

aka half the chord, is the square root of (17² – 8²).  

Half the chord is sqrt(225) which is 15.  

Therefore the whole chord is 30.  

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The quadratic equation $x^2-5x+5=-4x+7$ has two roots. Find the absolute difference of these roots.  

                                         x^2-5x+5=-4x+7  

Combine like terms           x² – x – 2  =  0  

Factor the quadratic         (x – 2)(x + 1)  =  0  

So, the roots are                 x  =  2  

                                           x  =  –1   

Can you take it from there?  

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