Algebra

 

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

 

                                                              5x² – kx + 8 – 2x² + 25 = 0    

Combine like terms and arrange in    

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

The discriminant of a quadratic equation written in standard form is b² – 4ac.  

 

For the equation to have imaginary solutions, its discriminant has to be negative.   

 

so [ (–k)² – (4)(3)(33) ] has to be negative ... that is, k² has to be less than 396    

 

The square root of 396 is ≈ 19.9   

 

The largest integer less than 19.9 is 19, so k = 19 is the largest integer that will produce an imaginary root.    

 

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