Quadratic formula¶
An quadratic equation $\( ax^2 + bx + c = 0 \)$
Is solved by $\( x = \frac{-b \plusmn \sqrt{b^2 - 4ac}}{2a} \)$
\(a \ge 0\)
In general this has two roots (real/complex).
\(b^2-4ac = 0\) the two roots are equal and real
\(b^2 \ge 4ac\) then the two roots are unequal and real
\(b^2 \le 4ac\) the two roots are unequal and complex, thus has a mathematical solution but no physical
Prof¶
\[\begin{split}
ax^2 + bx + c = 0 \\
4a^2 + 4abx + 4ac = 0 \\
4a^2x^2 + 4abx = -4ac \\
4a^2x^2 + 4abx+ b^2 = b^2 -4ac \\
(2ax + b)^2 = b^2 - 4ac \\
2ax + b = \plusmn \sqrt{b^2 - 4ac} \\
2ax = - b \plusmn \sqrt{b^2 - 4ac} \\
x = \frac{- b \plusmn \sqrt{b^2 - 4ac}}{2a}
\end{split}\]