Solution to Quadratic Equation
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Theorem
The quadratic equation of the form $a x^2 + b x + c = 0$ has solutions:
- $x = \dfrac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}$
Real Coefficients
Let $a, b, c \in \R$.
The quadratic equation $a x^2 + b x + c = 0$ has:
- Two real solutions if $b^2 - 4 a c > 0$
- One real solution if $b^2 - 4 a c = 0$
- Two complex solutions if $b^2 - 4 a c < 0$, and those two solutions are complex conjugates.
Proof
\(\ds a x^2 + b x + c\) | \(=\) | \(\ds 0\) | by hypothesis | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\paren {2 a x + b}^2 - b^2 + 4 a c} {4 a}\) | \(=\) | \(\ds 0\) | Completing the Square | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds \paren {2 a x + b}^2\) | \(=\) | \(\ds b^2 - 4 a c\) | simplification | ||||||||||
\(\ds \leadsto \ \ \) | \(\ds x\) | \(=\) | \(\ds \frac {-b \pm \sqrt {b^2 - 4 a c} } {2 a}\) | solving for $x$ |
$\blacksquare$
Also known as
Solution to Quadratic Equation is often referred to as the quadratic formula.
Sources
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- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): quadratic equation