What does \(\displaystyle x = \dfrac{-\ b \pm \sqrt{b^2 - 4ac}}{2a}\) mean exactly? It means
\(\displaystyle x = \dfrac{-\ b + \sqrt{b^2 - 4ac}}{2a} \text { OR } x = \dfrac{-\ b - \sqrt{b^2 - 4ac}}{2a}.\)
Now \(\displaystyle x = \dfrac{-\ b \pm \sqrt{b^2 - 4ac}}{2a} = \dfrac{(-\ 1)(-\ b \pm \sqrt{b^2 - 4 ac})}{(-\ 1)(2a)} = \dfrac{b \pm (-\ \sqrt{b^2 -4ac})}{-\ 2a}\implies\)
\(\displaystyle x = \dfrac{b + (-\ \sqrt{b^2 - 4 ac})}{-\ 2a} \text { OR } x = \dfrac{b - (-\ \sqrt{b^2 - 4ac})}{-\ 2a} \implies\)
\(\displaystyle x = \dfrac{b - \sqrt{b^2 - 4ac}}{-\ 2a} \text { OR } \dfrac{b + \sqrt{b^2 - 4ac}}{-\ 2a} \implies\)
\(\displaystyle x = \dfrac{b + \sqrt{b^2 - 4ac}}{-\ 2a} \text { OR } \dfrac{b - \sqrt{b^2 - 4ac}}{-\ 2a} \implies\)
\(\displaystyle x = \dfrac{b \pm \sqrt{b^2 - 4ac}}{-\ 2a}.\)
It comes from the definition of \(\displaystyle \pm\) as plus OR minus and the reversibility of "or." The two formulas mean EXACTLY THE SAME THING as tkhunny correctly pointed out.
The plus-minus sign is a mathematical symbol with multiple meanings across mathematics and non-mathematics areas. And, it is used in different ways within mathematics itself,
sometimes, depending on which topic/problem in mathematics one is covering. But within this context of the Quadratic Formula here, the plus-minus symbol means one thing:
"plus or minus."
.
https://en.wikipedia.org/wiki/Plus-minus_sign
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