If the quadratic formula is used to find the roots of the equation $x^{2} - 6 x - 19 = 0$ , what are the correct roots.

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Solution

Finding correct roots of quadratic equation via quadratic formula :
As we know, for a quadratic equation $a x^{2} + b x + c = 0$ the two roots $(α, β)$ can be given as
$(α, β) = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
For the equation $x^{2} - 6 x - 19 = 0$ , the values of $a, b, c$ are $1, - 6, - 19$ respectively.
Now using the quadratic formula to get roots.
$(α, β) = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
$\Rightarrow (α, β) = \frac{6 \pm \sqrt{6^{2} - 4 \times 1 \times (- 19)}}{2 \times 1} (Substitutedthevalues) = \frac{6 \pm \sqrt{36 + 76}}{2} = \frac{6 \pm \sqrt{112}}{2} = \frac{6 \pm 4 \sqrt{7}}{2} = 3 \pm 2 \sqrt{7}$
Hence, the roots of the given quadratic equation is $3 + 2 \sqrt{7}$ and $3 - 2 \sqrt{7}$ .

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