Lösung 2.3:4c
Aus Online Mathematik Brückenkurs 1
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| - | The equation | + | 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. |
| - | <math> | + | |
| - | is a second-degree equation which has | + | |
| - | <math>x= | + | |
| - | and | + | |
| - | <math>x=\sqrt | + | |
| - | as roots; when | + | |
| - | <math>x= | + | |
| - | <math>x=\sqrt | + | |
| - | the second factor is zero. | + | |
| - | If we expand the | + | 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>}} | ||
| - | <math>\begin{align} | ||
| - | & \left( x-\text{3} \right)\left( x-\sqrt{\text{3}} \right)=x^{2}-\sqrt{\text{3}}x-3x+3\sqrt{\text{3}} \\ | ||
| - | & =x^{2}-\left( 3+\sqrt{\text{3}} \right)x+3\sqrt{\text{3}}=0 \\ | ||
| - | \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. | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | where | + | |
| - | <math>a\ne 0</math> | + | |
| - | is a constant. | + | |
Version vom 10:39, 29. Sep. 2008
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,
Note: the general answer is
where \displaystyle a\ne 0 is a constant.
