Writing an Equation in Vertex
Form by Completing the Square
We want to write an equation in vertex form y  ax  h   k , by using a process
called completing the square. We will start with the standard form of the
quadratic equation, y  ax 2  bx  c and then rewrite the equation in vertex form.
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EXAMPLE #1
Write y  x 2  20 x  9 in vertex from.
1. Add 9 to both sides, so the constant is eliminated from the right hand side.
y  9  x 2  20 x  9  9
y  9  x 2  20 x
2. Now complete the square on the right hand side.
2


2
1 
1

2
Since b  20 , then  b  =   20  = 10  = 100
2 
2

So completing the square, we must add 100 to both sides, because
we are dealing with an equation.
y  9  100  x 2  20 x  100
y  109  x 2  20 x  100
3. We also know how to write the right side of the equation in factored form.

Again, we know b  20 , so
1


y  109   x   20 
2


2
y  109  x  10
2
4. Now the equation is written in vertex form.

y   x  10  109 , with a vertex of ( -10, -109 )
Algebra 2 Chapter 1.7
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EXAMPLE #2
Write y  x 2  6 x  7 in vertex from.
1. Subtract 7 to both sides, so the constant is eliminated from the right hand
side.
y  7  x 2  6x  7  7
y  7  x 2  6x
2. Now complete the square on the right hand side.
2
2
2
36
1 
1
  6
a. Since b  6 , then  b  =   6  =    =
=9
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2
  2
2 
b. So completing the square, we must add 9 to both sides, because
we are dealing with an equation.
y  7  9  x 2  6x  9
y  2  x 2  6x  9
3. We also know how to write the right side of the equation in factored form.

Again, we know b  6 , so
1


y  2   x   6 
2


y  2   x  3
2
2
4. Now the equation is written in vertex form.

y  x  3  2 , with a vertex of 3,2
Algebra 2 Chapter 1.7
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