SOLUTION: Write the quadratic equation in vertex form. What is the vertex? y=3x^2+30x+68

Algebra ->  College  -> Linear Algebra -> SOLUTION: Write the quadratic equation in vertex form. What is the vertex? y=3x^2+30x+68      Log On


   

Question 117607: Write the quadratic equation in vertex form. What is the vertex?
y=3x^2+30x+68
Found 2 solutions by MathLover1, stanbon:
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

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


y=3+x%5E2%2B30+x%2B68 Start with the given equation



y-68=3+x%5E2%2B30+x Subtract 68 from both sides



y-68=3%28x%5E2%2B10x%29 Factor out the leading coefficient 3



Take half of the x coefficient 10 to get 5 (ie %281%2F2%29%2810%29=5).


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





y-68=3%28x%5E2%2B10x%2B25-25%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 25 does not change the equation




y-68=3%28%28x%2B5%29%5E2-25%29 Now factor x%5E2%2B10x%2B25 to get %28x%2B5%29%5E2



y-68=3%28x%2B5%29%5E2-3%2825%29 Distribute



y-68=3%28x%2B5%29%5E2-75 Multiply



y=3%28x%2B5%29%5E2-75%2B68 Now add 68 to both sides to isolate y



y=3%28x%2B5%29%5E2-7 Combine like terms




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




Check:


Notice if we graph the original equation y=3x%5E2%2B30x%2B68 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C3x%5E2%2B30x%2B68%29 Graph of y=3x%5E2%2B30x%2B68. Notice how the vertex is (-5,-7).



Notice if we graph the final equation y=3%28x%2B5%29%5E2-7 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C3%28x%2B5%29%5E2-7%29 Graph of y=3%28x%2B5%29%5E2-7. Notice how the vertex is also (-5,-7).



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

Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
Write the quadratic equation in vertex form. What is the vertex?
y=3x^2+30x+68
---------
This equation graphs as a parabola; it has a highest point or
a lowest point which is called the vertex.
---------------
Put the equation in vertex form by completing the square on the x-terms.
--------------
3(x^2+10x +?) = y-68+3*?
3(x^2+10x+5^2) = y-68+3*25
3(x+5)^2 = y+7
3(x+5)^2 = (y+7)
(x+5)^2 = (1/3)(y+7)
---------------
Vertex at (-5,-7)
-----------------
Graph:
graph%28400%2C300%2C-10%2C10%2C-10%2C70%2C3x%5E2%2B30x%2B68%29
====================
Cheers,
Stan H.