Completing The Square

Completing the Square - Problem Solving


Consider two functions f(x)=4x2+14x+4g(x)=7x2+8x+3.\begin{aligned} f(x) &= -4x^2 + 14 x + 4 \\ g(x) &= -7 x^2 + 8 x + 3. \end{aligned} Let a a be the maximum value of f(x)+g(x), f(x) + g(x) , and bb the value of xx at that moment. Then what is the value of a+b? a+b?

For real numbers xx and yy, the polynomial x2+2y2+2xy8x10y+30x^2+2y^2+2xy-8x-10y+30 has a minimum value mm at x=ax=a and y=by=b. What is the value of abmabm?

Let f(x)=5x+60f(x)= 5 x + 60 and g(x)=4x+80.g(x)= 4 x + 80. If aa is the minimum value of f(x)×g(x) f(x) \times g(x) and bb is the value of xx at that moment, what is the value of ba? b-a?

Consider two points A=(2,5)A=(2,5) and B=(4,4)B=(4,4), and a third point P=(a,0)P=(a, 0) on the xx-axis. If the minimum value of PA2+PB2\overline{PA}^2+\overline{PB}^2 is cc, what is a+ca+c?

Consider the parabola y=x2+k4+p42k2x2p2x+2k2p2+36,y = x^2 + k^4 + p^4 -2k^2 x -2p^2x + 2k^2p^2 + 36, where kk and pp are parameters. If the vertex of the parabola lies on the line y=125x, y = \frac{ 1 }{25} x, what is the value of k2+p2? k^2 + p^2?


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