document.write( "Question 7495: Determine the vertex of y=x^2-8x+22. \n" ); document.write( "
Algebra.Com's Answer #4139 by ichudov(507)\"\" \"About 
You can put this solution on YOUR website!
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "\n" ); document.write( "
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation \"ax%5E2%2Bbx%2Bc=0\" (in our case \"1x%5E2%2B-8x%2B22+=+0\") has the following solutons:
\n" ); document.write( "
\n" ); document.write( " \"x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca\"
\n" ); document.write( "
\n" ); document.write( " For these solutions to exist, the discriminant \"b%5E2-4ac\" should not be a negative number.
\n" ); document.write( "
\n" ); document.write( " First, we need to compute the discriminant \"b%5E2-4ac\": \"b%5E2-4ac=%28-8%29%5E2-4%2A1%2A22=-24\".
\n" ); document.write( "
\n" ); document.write( " The discriminant -24 is less than zero. That means that there are no solutions among real numbers.

\n" ); document.write( " If you are a student of advanced school algebra and are aware about imaginary numbers, read on.

\n" ); document.write( "
\n" ); document.write( " In the field of imaginary numbers, the square root of -24 is + or - \"sqrt%28+24%29+=+4.89897948556636\".
\n" ); document.write( "
\n" ); document.write( " The solution is
\n" ); document.write( "
\n" ); document.write( " Here's your graph:
\n" ); document.write( "\"graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B22+%29\"

\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "the vertex is at the midpoint between roots (never mind that roots are complex, you can still add them and divide sum by half and get a real number). That midpoint is x=4. Calculate the value of x^2-8x+22 for x=4, and you would get the x and y value this way. \r
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "
Solved by pluggable solver: SUBSTITUTION solver -- compute expression given values
Your Result:

\n" ); document.write( "
\n" ); document.write( "

\n" ); document.write( " YOUR ANSWER\n" ); document.write( "


\n" ); document.write( "\n" ); document.write( "
    \n" ); document.write( "
  • Graphical form: \"x%5E2-8x%2B22\" simplifies to \"6\"
  • Text form: x^2-8x+22 simplifies to 6
  • Cartoon (animation) form: \"simplify_cartoon%28+x%5E2-8x%2B22+%29\"
    For tutors: simplify_cartoon( x^2-8x+22 )
    • \n" ); document.write( "
  • If you have a website, here's a link to this solution.
    • \n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "

DETAILED EXPLANATION

\n" ); document.write( "
Look at \"highlight_red%28+x+%29%5E2-8%2Ax%2B22\".
Replaced x with 4
It becomes \"highlight_green%28+4+%29%5E2-8%2Ax%2B22\".
\n" ); document.write( "
Look at \"highlight_red%28+4+%29%5Ehighlight_red%28+2+%29-8%2Ax%2B22\".
Replace constants in factor: \"highlight_red%28+4+%29\",\"highlight_red%28+2+%29\" with \"highlight_green%28+16+%29\"
It becomes \"highlight_green%28+16+%29-8%2Ax%2B22\".
\n" ); document.write( "
Look at \"16-8%2Ahighlight_red%28+x+%29%2B22\".
Replaced x with 4
It becomes \"16-8%2Ahighlight_green%28+4+%29%2B22\".
\n" ); document.write( "
Look at \"highlight_red%28+16+%29-8%2A4%2Bhighlight_red%28+22+%29\".
Added fractions or integers together
It becomes \"highlight_green%28+38+%29-8%2A4\".
\n" ); document.write( "
Look at \"38-highlight_red%28+8+%29%2Ahighlight_red%28+4+%29\".
Multiplied numerator integers
It becomes \"38-highlight_green%28+32+%29\".
\n" ); document.write( "
Look at \"highlight_red%28+38+%29-highlight_red%28+32+%29\".
Added fractions or integers together
It becomes \"highlight_green%28+6+%29\".
Result: \"6\"
\n" ); document.write( "

Universal Simplifier and Solver

\n" ); document.write( "
Done!

\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Answer: x=4, y=6 \n" ); document.write( "
\n" );