Lösung 3.2:5

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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After squaring both sides, we obtain the equation
After squaring both sides, we obtain the equation
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{{Displayed math||<math>3x-2 = (2-x)^2</math>|(*)}}
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{{Abgesetzte Formel||<math>3x-2 = (2-x)^2</math>|(*)}}
and if we expand the right-hand side and then collect the terms, we get
and if we expand the right-hand side and then collect the terms, we get
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{{Displayed math||<math>x^{2}-7x+6=0\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>x^{2}-7x+6=0\,\textrm{.}</math>}}
Completing the square of the left-hand side, we obtain
Completing the square of the left-hand side, we obtain
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
x^{2}-7x+6
x^{2}-7x+6
&= \Bigl(x-\frac{7}{2}\Bigr)^2 - \Bigl(\frac{7}{2}\Bigr)^2+6\\[5pt]
&= \Bigl(x-\frac{7}{2}\Bigr)^2 - \Bigl(\frac{7}{2}\Bigr)^2+6\\[5pt]
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which means that the equation can be written as
which means that the equation can be written as
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{{Displayed math||<math>\Bigl(x-\frac{7}{2}\Bigr)^2 = \frac{25}{4}</math>}}
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{{Abgesetzte Formel||<math>\Bigl(x-\frac{7}{2}\Bigr)^2 = \frac{25}{4}</math>}}
and the solutions are therefore
and the solutions are therefore

Version vom 08:41, 22. Okt. 2008

After squaring both sides, we obtain the equation

\displaystyle 3x-2 = (2-x)^2 (*)

and if we expand the right-hand side and then collect the terms, we get

\displaystyle x^{2}-7x+6=0\,\textrm{.}

Completing the square of the left-hand side, we obtain

\displaystyle \begin{align}

x^{2}-7x+6 &= \Bigl(x-\frac{7}{2}\Bigr)^2 - \Bigl(\frac{7}{2}\Bigr)^2+6\\[5pt] &= \Bigl(x-\frac{7}{2}\Bigr)^2 - \frac{49}{4} + \frac{24}{4}\\[5pt] &= \Bigl(x-\frac{7}{2}\Bigr)^2 - \frac{25}{4} \end{align}

which means that the equation can be written as

\displaystyle \Bigl(x-\frac{7}{2}\Bigr)^2 = \frac{25}{4}

and the solutions are therefore

  • \displaystyle x = \frac{7}{2} + \sqrt{\frac{25}{4}} = \frac{7}{2} + \frac{5}{2} = \frac{12}{2} = 6\,,
  • \displaystyle x = \frac{7}{2} - \sqrt{\frac{25}{4}} = \frac{7}{2} - \frac{5}{2} = \frac{2}{2} = 1\,\textrm{.}

Substituting \displaystyle x=1 and \displaystyle x=6 into the quadratic equation (*) shows that we have solved the equation correctly.

  • x = 1: \displaystyle \ \text{LHS} = 3\cdot 1-2 = 1\ and \displaystyle \ \text{RHS} = (2-1)^2 = 1
  • x = 6: \displaystyle \ \text{LHS} = 3\cdot 6-2 = 16\ and \displaystyle \ \text{RHS} = (2-6)^2 = 16

Finally, we need to sort away possible spurious roots to the root equation by verifying the solutions.

  • x = 1: \displaystyle \ \text{LHS} = \sqrt{3\cdot 1-2} = 1\ and \displaystyle \ \text{RHS} = 2-1 = 1
  • x = 6: \displaystyle \ \text{LHS} = \sqrt{3\cdot 6-2} = 4\ and \displaystyle \ \text{RHS} = 2-6 = -4

This shows that the root equation has the solution \displaystyle x=1\,.

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