SOLUTION: Determine the vertex of y=x^2-8x+22.

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Question 7495: Determine the vertex of y=x^2-8x+22.
Answer by ichudov(507) About Me  (Show Source):
You can put this solution on YOUR website!
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:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-8%29%5E2-4%2A1%2A22=-24.

The discriminant -24 is less than zero. That means that there are no solutions among real numbers.

If you are a student of advanced school algebra and are aware about imaginary numbers, read on.


In the field of imaginary numbers, the square root of -24 is + or - sqrt%28+24%29+=+4.89897948556636.

The solution is

Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-8%2Ax%2B22+%29


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.
Solved by pluggable solver: SUBSTITUTION solver -- compute expression given values
Your Result:


YOUR ANSWER


  • 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 )
  • If you have a website, here's a link to this solution.

DETAILED EXPLANATION

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.
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.
Look at 16-8%2Ahighlight_red%28+x+%29%2B22.
Replaced x with 4
It becomes 16-8%2Ahighlight_green%28+4+%29%2B22.
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.
Look at 38-highlight_red%28+8+%29%2Ahighlight_red%28+4+%29.
Multiplied numerator integers
It becomes 38-highlight_green%28+32+%29.
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

Universal Simplifier and Solver


Done!


Answer: x=4, y=6