Equating Coefficients/Examples/Quadratic
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Example of Equating Coefficients
Consider the quadratic equation:
- $(1): \quad x^2 + a x + b = 0$
Let the roots of $(1)$ be $\alpha$ and $\beta$.
Thus:
\(\ds x^2 + a x + b\) | \(=\) | \(\ds \paren {x - \alpha} \paren {x - \beta}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds x^2 - \paren {\alpha + \beta} x + \alpha \beta\) |
Hence by equating coefficients:
\(\ds -a\) | \(=\) | \(\ds \alpha + \beta\) | as the coefficients of $x^1$ must be the same | |||||||||||
\(\ds b\) | \(=\) | \(\ds \alpha \beta\) | as the coefficients of $x^0$ must be the same |
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): equate coefficients
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): equate coefficients