There are three things we need to consider as a pre-amble before we start.
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color(blue)("Point 1")
Consider (3x)^2 Inside the brackets the coefficient is presented as 3. Outside the bracket it has been squared so it will be 9 in that:
9xx(x)^2=(3x)^2 another example ->" "16xx(x)^2=(4x)^2
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color(blue)("Point 2")
1/3xx(3x-15)^2 =((3x)/(sqrt(3))-15/sqrt(3))^2
so 1/9(3x-15)^2=((3x)/3-15/3)^2
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color(blue)("Point 3")
To convert the given equation into vertex form we need to end up with the format of:
y=a(x-b/(2a))^2 +c" " where b can be positive or negative.
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color(blue)("Solving your question")
With the format of the given question you are already part way to building the vertex equation format of completing the square. So this is what I am going to do.
Given:" " y= (1/6)(3x-15)^2-31
To remove the coefficient of x within the brackets multiply the bracketed part by 1, but in the form of color(blue)(9/9)
y= color(blue)(9/9)(1/6)(3x-15)^2-31
y=(color(blue)(9))/6((3x)/(color(blue)(3))-15/(color(blue)(3)))^2-31
y=9/6(x-5)^2-31" "color(brown)("This is vertex form")
![Tony B]()
Tony B
Thus:
x_("vertex")=(-1)xx(-5)=5
y_("vertex") = -31 Notice that this is the value of the constant c
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Vertex" "=" "(x,y)" "->" "(5,-31)
