Solution 2.3:9c

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Current revision (14:16, 29 September 2008) (edit) (undo)
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To determine all the points on the curve
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To determine all the points on the curve <math>y=3x^{2}-12x+9</math> which also lie on the ''x''-axis we substitute the equation of the ''x''-axis i.e. <math>y=0</math> in the equation of the curve and obtain that ''x'' must satisfy
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<math>y=3x^{2}-12x+9</math>
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which also lie on the
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<math>x</math>
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-axis we substitute the equation of the
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<math>x</math>
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-axis i.e.
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<math>y=0</math>
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in the equation of the curve and obtain that
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<math>x</math>
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must satisfy
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{{Displayed math||<math>0 = 3x^{2}-12x+9\,\textrm{.}</math>}}
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<math>3x^{2}-12x+9=0</math>
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After dividing by 3 and completing the square the right-hand side is
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{{Displayed math||<math>x^{2}-4x+3 = (x-2)^{2} - 2^{2} + 3 = (x-2)^{2} - 1</math>}}
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After dividing by
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and thus the equation has solutions <math>x=2\pm 1,</math>
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<math>3</math>
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i.e. <math>x=2-1=1</math> and <math>x=2+1=3\,</math>.
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and completing the square the right-hand side is
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<math>x^{2}-4x+3=\left( x-2 \right)^{2}-2^{2}+3=\left( x-2 \right)^{2}-1</math>
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and thus the equation has solutions
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<math>x=2\pm 1,</math>
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i.e.
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<math>x=2-1=1</math>
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and
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<math>x=2+1=3.</math>
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The points where the curve cut the
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<math>x</math>
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-axis are
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<math>\left( 1 \right.,\left. 0 \right)</math>
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and
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<math>\left( 3 \right.,\left. 0 \right)</math>
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The points where the curve cut the ''x''-axis are (1,0) and (3,0).
[[Image:2_3_9_c.gif|center]]
[[Image:2_3_9_c.gif|center]]

Current revision

To determine all the points on the curve \displaystyle y=3x^{2}-12x+9 which also lie on the x-axis we substitute the equation of the x-axis i.e. \displaystyle y=0 in the equation of the curve and obtain that x must satisfy

\displaystyle 0 = 3x^{2}-12x+9\,\textrm{.}

After dividing by 3 and completing the square the right-hand side is

\displaystyle x^{2}-4x+3 = (x-2)^{2} - 2^{2} + 3 = (x-2)^{2} - 1

and thus the equation has solutions \displaystyle x=2\pm 1, i.e. \displaystyle x=2-1=1 and \displaystyle x=2+1=3\,.

The points where the curve cut the x-axis are (1,0) and (3,0).


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