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Completing The Square

Learn how to eliminate the linear term and see through messy quadratics. You’ll know just how high to shoot your three-pointer to avoid tall defenders.

Completing the Square - Applications


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

As \(x\) ranges over all real values, what is the smallest integer value of

\[ \frac{ x^2 + 22x + 333} { 11} ? \]

Details and assumptions

Note that we are not asking for the value of \(x\).

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

The quadratic equation \(3x^2+24x-8=0\) can be expressed as \[(x-m)^2=n,\] where \(m\) and \(n\) are constants. What is the value of \(m+3n\)?

If the solution of the equation \[x^2+6x-1=0\] is \(x=a \pm \sqrt{b}\), what is \(a+b\)?


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