Solution 2.3:4c
From Förberedande kurs i matematik 1
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- | {{ | + | The equation <math>(x-3)(x-\sqrt{3}\,)=0</math> is a second-degree equation which has <math>x=3</math> and <math>x=\sqrt{3}</math> as roots; when <math>x=3</math>, the first factor is zero and when <math>x=\sqrt{3}</math> the second factor is zero. |
- | < | + | |
- | {{ | + | If we expand the equation's left-hand side, we get the equation in standard form, |
+ | |||
+ | {{Displayed math||<math>\begin{align} | ||
+ | (x-3)(x-\sqrt{3}\,) | ||
+ | &= x^{2}-\sqrt{3}x-3x+3\sqrt{3}\\[5pt] | ||
+ | &= x^{2}-(3+\sqrt{3}\,)x+3\sqrt{3}=0\,\textrm{.} | ||
+ | \end{align}</math>}} | ||
+ | |||
+ | |||
+ | Note: the general answer is | ||
+ | |||
+ | {{Displayed math||<math>ax^{2}-(3+\sqrt{3}\,)ax+3\sqrt{3}a=0\,,</math>}} | ||
+ | |||
+ | where <math>a\ne 0</math> is a constant. |
Current revision
The equation \displaystyle (x-3)(x-\sqrt{3}\,)=0 is a second-degree equation which has \displaystyle x=3 and \displaystyle x=\sqrt{3} as roots; when \displaystyle x=3, the first factor is zero and when \displaystyle x=\sqrt{3} the second factor is zero.
If we expand the equation's left-hand side, we get the equation in standard form,
\displaystyle \begin{align}
(x-3)(x-\sqrt{3}\,) &= x^{2}-\sqrt{3}x-3x+3\sqrt{3}\\[5pt] &= x^{2}-(3+\sqrt{3}\,)x+3\sqrt{3}=0\,\textrm{.} \end{align} |
Note: the general answer is
\displaystyle ax^{2}-(3+\sqrt{3}\,)ax+3\sqrt{3}a=0\,, |
where \displaystyle a\ne 0 is a constant.