SOLUTION: Complete the square to write each function in the form {{{f(x)=a(x-h)^2+k}}} {{{f(x)=2x^2-5x+10}}}

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Complete the square to write each function in the form {{{f(x)=a(x-h)^2+k}}} {{{f(x)=2x^2-5x+10}}}       Log On


   

Question 747983: Complete the square to write each function in the form f%28x%29=a%28x-h%29%5E2%2Bk
f%28x%29=2x%5E2-5x%2B10
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!
f%28x%29=2x%5E2-5x%2B10
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=2+x%5E2%2B5+x%2B10 Start with the given equation



y-10=2+x%5E2%2B5+x Subtract 10 from both sides



y-10=2%28x%5E2%2B%285%2F2%29x%29 Factor out the leading coefficient 2



Take half of the x coefficient 5%2F2 to get 5%2F4 (ie %281%2F2%29%285%2F2%29=5%2F4).


Now square 5%2F4 to get 25%2F16 (ie %285%2F4%29%5E2=%285%2F4%29%285%2F4%29=25%2F16)





y-10=2%28x%5E2%2B%285%2F2%29x%2B25%2F16-25%2F16%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 25%2F16 does not change the equation




y-10=2%28%28x%2B5%2F4%29%5E2-25%2F16%29 Now factor x%5E2%2B%285%2F2%29x%2B25%2F16 to get %28x%2B5%2F4%29%5E2



y-10=2%28x%2B5%2F4%29%5E2-2%2825%2F16%29 Distribute



y-10=2%28x%2B5%2F4%29%5E2-25%2F8 Multiply



y=2%28x%2B5%2F4%29%5E2-25%2F8%2B10 Now add 10 to both sides to isolate y



y=2%28x%2B5%2F4%29%5E2%2B55%2F8 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=2, h=-5%2F4, and k=55%2F8. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=2x%5E2%2B5x%2B10 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2x%5E2%2B5x%2B10%29 Graph of y=2x%5E2%2B5x%2B10. Notice how the vertex is (-5%2F4,55%2F8).



Notice if we graph the final equation y=2%28x%2B5%2F4%29%5E2%2B55%2F8 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C2%28x%2B5%2F4%29%5E2%2B55%2F8%29 Graph of y=2%28x%2B5%2F4%29%5E2%2B55%2F8. Notice how the vertex is also (-5%2F4,55%2F8).



So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.