SOLUTION: -3x^2+7x=-5 solve by completing the square. then write in vertex form

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Question 124681: -3x^2+7x=-5 solve by completing the square. then write in vertex form
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

-3x%5E2%2B7x=-5
-3x%5E2%2B7x+%2B+5+=+0

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-3+x%5E2%2B7+x%2B5 Start with the given equation



y-5=-3+x%5E2%2B7+x Subtract 5 from both sides



y-5=-3%28x%5E2%2B%28-7%2F3%29x%29 Factor out the leading coefficient -3



Take half of the x coefficient -7%2F3 to get -7%2F6 (ie %281%2F2%29%28-7%2F3%29=-7%2F6).


Now square -7%2F6 to get 49%2F36 (ie %28-7%2F6%29%5E2=%28-7%2F6%29%28-7%2F6%29=49%2F36)





y-5=-3%28x%5E2%2B%28-7%2F3%29x%2B49%2F36-49%2F36%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 49%2F36 does not change the equation




y-5=-3%28%28x-7%2F6%29%5E2-49%2F36%29 Now factor x%5E2%2B%28-7%2F3%29x%2B49%2F36 to get %28x-7%2F6%29%5E2



y-5=-3%28x-7%2F6%29%5E2%2B3%2849%2F36%29 Distribute



y-5=-3%28x-7%2F6%29%5E2%2B49%2F12 Multiply



y=-3%28x-7%2F6%29%5E2%2B49%2F12%2B5 Now add 5 to both sides to isolate y



y=-3%28x-7%2F6%29%5E2%2B109%2F12 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-3, h=7%2F6, and k=109%2F12. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-3x%5E2%2B7x%2B5 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3x%5E2%2B7x%2B5%29 Graph of y=-3x%5E2%2B7x%2B5. Notice how the vertex is (7%2F6,109%2F12).



Notice if we graph the final equation y=-3%28x-7%2F6%29%5E2%2B109%2F12 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-3%28x-7%2F6%29%5E2%2B109%2F12%29 Graph of y=-3%28x-7%2F6%29%5E2%2B109%2F12. Notice how the vertex is also (7%2F6,109%2F12).



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