Question

slove 3x2+2x+3 by using formula
$3x {}^{2} + 2x + 3 \: by \: \: using \: formula$
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Answer 1

---------- QUADRATIC FORMULA FOR $ax^2+2b'x+c=0$ ----------

$2b'$ stands for the 'even number coefficient' of the 'linear term'.

Apply quadratic formula first.

$x=\frac{-2b'\pm\sqrt{4b'^2-4ac} }{2a}$

$=\frac{-2b'\pm 2\sqrt{b'^2-ac} }{2a}$

$=\frac{-b'\pm \sqrt{b'2-ac} }{a}$

∴ $x=\frac{-b'\pm \sqrt{b'^2-ac} }{a}$ are the roots of a quadratic equation $ax^2+2b'x+c=0$.

------------------------------ Find the roots of $2x^2+2x+3$ ------------------------------

Use the formula we've proved before.

$x=\frac{-1\pm\sqrt{1^2-3\times 3} }{3}=\frac{-1\pm\sqrt{-8} }{3}=\frac{-1\pm 2\sqrt{2}i }{3}$

The roots are $\frac{-1\pm 2\sqrt{2}i }{3}$.