Algebra

Completing The Square

Completing the Square - Applications

         

As xx ranges over all real values, what is the minimum value of x2+28x+313 x^ 2 + 28 x + 313 ?

As xx ranges over all real values, what is the smallest integer value of

x2+22x+33311? \frac{ x^2 + 22x + 333} { 11} ?

Details and assumptions

Note that we are not asking for the value of xx.

If 36+165=a+bc \sqrt{ 36 + 16 \sqrt{5} } = a + b \sqrt{c} , where a,ba, b and cc are positive integers and cc is not divisible by the square of a prime. Determine a+b+ca+b+c.

The quadratic equation 3x2+24x8=03x^2+24x-8=0 can be expressed as (xm)2=n,(x-m)^2=n, where mm and nn are constants. What is the value of m+3nm+3n?

If the solution of the equation x2+6x1=0x^2+6x-1=0 is x=a±bx=a \pm \sqrt{b}, what is a+ba+b?

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