Question 520709
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17x² – 10xy + 2y² – 6x + 2 = 0
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Rearrange the terms in descending order of y
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2y² - 10xy + 17x² - 6x + 2 = 0
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Think of it like this:
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2y² + (-10x)y + (17x² - 6x + 2) = 0 
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and use the quadratic formula
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with a = 2, b = -10x and c = 17x² - 6x + 2
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+&x=0003 ALIGN=MIDDLE  ALT="%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+"  >
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%28-%28-10x%29+%2B-+sqrt%28+%28-10x%29%5E2-4%2A%282%29%2A%2817x%5E2-6x%2B2%29+%29%29%2F%282%2A%282%29%29+&x=0003 ALIGN=MIDDLE  ALT="%28-%28-10x%29+%2B-+sqrt%28+%28-10x%29%5E2-4%2A%282%29%2A%2817x%5E2-6x%2B2%29+%29%29%2F%282%2A%282%29%29+"  >
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%2810x+%2B-+sqrt%28100x%5E2-8%2817x%5E2-6x%2B2%29+%29%29%2F4+&x=0003 ALIGN=MIDDLE  ALT="%2810x+%2B-+sqrt%28100x%5E2-8%2817x%5E2-6x%2B2%29+%29%29%2F4+"  >
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%2810x+%2B-+sqrt%28100x%5E2-136x%5E2%2B48x-16+%29%29%2F4+&x=0003 ALIGN=MIDDLE  ALT="%2810x+%2B-+sqrt%28100x%5E2-136x%5E2%2B48x-16+%29%29%2F4+"  >
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%2810x+%2B-+sqrt%28-36x%5E2%2B48x-16+%29%29%2F4+&x=0003 ALIGN=MIDDLE  ALT="%2810x+%2B-+sqrt%28-36x%5E2%2B48x-16+%29%29%2F4+"  >
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%2810x+%2B-+sqrt%28-4%289x%5E2-12x%2B4%29+%29%29%2F4+&x=0003 ALIGN=MIDDLE  ALT="%2810x+%2B-+sqrt%28-4%289x%5E2-12x%2B4%29+%29%29%2F4+"  >
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y = <IMG SRC=/cgi-bin/plot-formula.mpl?expression=%2810x+%2B-+sqrt%28-4%283x-2%29%283x-2%29+%29%29%2F4+&x=0003 ALIGN=MIDDLE  ALT="%2810x+%2B-+sqrt%28-4%283x-2%29%283x-2%29+%29%29%2F4+"  >
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y = {{{(10x +- sqrt(-4(3x-2)^2))/4 }}} 

y = {{{(10x +- 2i*sqrt((3x-2)^2))/4 }}} 

Divide every term in the top and bottom by 2

y = {{{(5x +- i*sqrt((3x-2)^2))/2 }}}

y = {{{(5x +- i*abs(3x-2))/2 }}}


That will be imaginary unless the imaginary part is 0, so

3x-2 = 0

   x = {{{2/3}}}

Substitute that

y = {{{(5(2/3))/2 }}} = {{{(10/3)/2}}} = {{{10/3}}}÷2 = {{{5/3}}}

and get y = {{{5/3}}}

Answer:  ({{{2/3}}},{{{5/3}}})

Edwin</pre>