Solution 2.3:4c

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

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