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Solving a Quadratic Equation by Completion of Squares Method

state conditi...

Question

state condition for which the roots of the equation $x^{2} + 2 (a - 1) x + a + 5 = 0$ are real.

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Solution

$\begin{matrix} x = \frac{- b \pm \sqrt{b^{2}} - 4 a c}{2 a} = \frac{- b}{2 a} \pm \frac{\sqrt{b^{2} - 4 a c}}{2 a} = \frac{- 2 (a - 1)}{2 \times 1} \pm \frac{\sqrt{{(2 (a - 1))}^{2} - 4 \times 1 \times (a + 5)}}{2} \pm \sqrt{{(a - 1)}^{2} - (a + 5)} {(a - 1)}^{2} > a + 5 \end{matrix}$
other wise no real root

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