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What is the largest value of k such that the equation 6x - x^2 = k has at least one real solution?

 Oct 25, 2022
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What is the largest value of k such that the equation 6x - x^2 = k has at least one real solution?  

 

 

Let's rearrange 6x - x^2 = k to standard quadratic equation form.   –x2 + 6x – k = 0  

 

Multiply both sides by –1 to get that pesky negative off the x2          x2 – 6x + k = 0  

 

What two numbers can we add together to get 6?  

There are 5 + 1,  4 + 2, and 3 + 3.  

What do we get when we multiply each of those pairs together?  

There are 5•1=5, 4•2=8, and 3•3=9.  

 

9 is the largest, so let's say that k = 9.  Now we have                        x2 – 6x + 9 = 0  

 

This will factor to                                                                                (x – 3)(x – 3)  

 Oct 25, 2022

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