Question 598070
Hi, there--
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Use the distance formula to find the distance between two points. The two different values of y will give you a point above and and a point below (2,70). 
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The distance formula is
{{{D=sqrt((x[2]-x[1])^2+(y[2]-y[1])^2)}}}
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Let (x[1],y[1])=(2,70).
Let (x[2],y[2])=(5,y).
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Substitute 5 for D and the known values for each point.
{{{D=sqrt((x[2]-x[1])^2+(y[2]-y[1])^2)}}}
{{{5=sqrt((5-2)^2+(y-70)^2)}}}
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Simplify and solve for y.
{{{5=sqrt((3)^2+(y-70)^2)}}}
{{{5=sqrt(9+y^2-140y+4900)}}}
{{{5=sqrt(y^2-140y+4909)}}}
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Square both sides of the equation to clear the square root.
{{{25=y^2-140y+4909}}}
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Set the polynomial equal to zero, and factor.
{{{y^2-140y+4884=0}}}
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We can use the quadratic formula or the factoring method to solve. 4884 looks like a friendly number to I would try to factor it.
 If I can't find factors right away, I switch to the quadratic formula. The Q.F. works for every quadratic, but not every quadratic is factorable.
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{{{484=2^2*3*11*37}}}; We want two factors with a sum of 140, and a product of 4884. I played around with the prime factors and got -2*3*11=-66 and -2*37=-74. So,
{{{y-74=0}}} OR {{{y-66=0}}}
{{{y=74}}} OR {{{y=66}}}
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Therefore, the two points are (5,66) and (5,74).
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Check your work for accuracy.
Distance between (2,70) and (5,66).
{{{D=sqrt((5-2)^2+(66-70)^2)}}}
{{{D=sqrt((3)^2+(-4)^2)}}}
{{{D=sqrt(9+16)}}}
{{{D=sqrt(25)}}}
{{{D=5}}}
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Distance between(2,70) and (5,74).
{{{D=sqrt((5-2)^2+(74-70)^2)}}}
{{{D=sqrt((3)^2+(4)^2)}}}
{{{D=sqrt(9+16)}}}
{{{D=sqrt(25)}}}
{{{D=5}}}
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That's it! Feel free to email via gmail if you have questions about the solution
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Ms.Figgy
math.in.the.vortex@gmail.com