Completing the Square Refresher
A quadratic equation in standard form is written y  ax 2  bx  c
A quadratic in vertex form is written y   x  h   k , where  h, k  is the vertex of the parabola.
2
To change an equation from standard form to vertex form, we complete the square
Write y  2 x 2  8x  5 in vertex form and find the vertex
Method 1
1.
Factor the leading coefficient (a) out of the first two terms
y  2  x2  4x   5
2.
Next, take half of the middle term (b) and square it, then add it within the parenthesis and subtract it
from outside the parenthesis – remember to distribute the a!




The b term is 4
Half of 4 is 2
2 squared is 4
Add 4 inside the parenthesis, subtract 2  4 from outside the parenthesis
y  2  x2  4x  4  5  8
2
3.
1 

Factor the parenthesis – it will always factor into  x  b  – and simplify the rest of the equation.
2 

y  2  x  2   3  vertex of the parabola is  2, 3
2
OVER 
Method 2
Write y  2 x 2  8x  5 in vertex form and find the vertex
1.
Move the c term to the other side of the equation
y  5  2 x2  8x
2.
Factor out the leading coefficient (a) and divide through by it.
y 5
  x2  4x
2 2
3.
Next, take half of the middle term (b) and square it, then add it to both sides




The b term is 4
Half of 4 is 2
2 squared is 4
Add 4 to both sides
y 5
  4  x2  4 x  4
2 2
2
4.
1 

Factor the right side of the equation – it will always factor into  x  b  – and then isolate y
2 

y 3
2
   x  2 
2 2
y
3
2
2
  x  2   y  2  x  2  3
2
2
Vertex of the parabola is  2, 3
Math 3
Spiral Review 4
Name__________________________________
1.
Find the vertex of the following equation by completing the square:
y  3x 2  24 x  11
2.
Solve the following equation by completing the square:
y  2 x 2  12 x  3
Math 3
Spiral Review 4
Name__________________________________
1.
Find the vertex of the following equation by completing the square:
y  3x 2  24 x  11
2.
Solve the following equation by completing the square:
y  2 x 2  12 x  3