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Completing the Square

Completing the square involves manipulating an expression so that it becomes a perfect square and we can factor it.

For example, if we have \(x^2 + 4x + 2 = 0\), we can make the expression on the left a perfect square by adding \( 2 \) to both sides ( because \( (x+2)^2 = x^2 + 4x + 4 \) ). This makes it significantly easier to solve the equation.

\[ \begin{align} x^2 + 4x + 2 &= 0 \\ x^2 + 4x + 4 &= 2 \\ (x+2)^2 &= 2 \\ x &= -2 \pm \sqrt{2} \end{align} \]

More generally,

\[ ax^2 + bx + c = a ( x + \frac{b}{2a} )^2 + \frac{ 4ac - b^2 } { 4a}. \]

Note by Arron Kau
3 years, 3 months ago

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