First slide

Theory of equations

Question

If ${ax}^{2} + bx + c = 0, then x =$

Moderate

A

$\frac{b \pm \sqrt{b^{2} - 4 ac}}{2 a}$
B

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

$\frac{2 c}{- b \pm \sqrt{b^{2} - 4 ac}}$
D

None of these

Solution

$x = \frac{- b \pm \sqrt{b^{2} - 4 ac}}{2 a}$
Since $\frac{2 c}{- b + \sqrt{b^{2} - 4 ac}} . \frac{- b - \sqrt{b^{2} - 4 ac}}{- b - \sqrt{b^{2} - 4 ac}}$
$= \frac{2 c (- b - \sqrt{b^{2} - 4 ac})}{4 ac} = \frac{- b - \sqrt{b^{2} - 4 ac}}{2 a}$
Similarly $\frac{2 c}{- b - \sqrt{b^{2} - 4 ac}} \times \frac{- b + \sqrt{b^{2} - 4 ac}}{- b + \sqrt{b^{2} - 4 ac}}$
$= \frac{2 c (- b + \sqrt{b^{2} - 4 ac})}{4 ac} = \frac{- b + \sqrt{b^{2} - 4 ac}}{2 a}$

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