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Find the constant $p$ such that $x^2 - 5x + p + 2x^2 - 8x$ is the square of a binomial.

 Nov 15, 2023
edited by bader  Nov 15, 2023
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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. 

.

 Nov 17, 2023

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