1. Completing the Square of x2 + bx + c
I.. Completing the square (getting the vertex form eq).
A) Turns y = x2 + bx + c into y = (x – h)2 + k.
1) Converts standard form into vertex form.
2) Quickly finds the vertex point (in vertex form).
a) Vertex is (h , k).
3) Can be solved for x to get the solutions (roots,
x-intercepts, etc.) without factoring.
a) Get the (x – h)2 part by itself.
b) Then square root both sides.
c) Solve for x.

2. Completing the Square of x2 + bx + c
II.. ½ the number, square the number. Move it on back.
A) Take ½ of the “b” term.
1) write it in (x )2 + c = y form.
B) Square the number you just wrote.
1) Write it on the other side of the = sign.
(x )2 + c = y [ a # squared is always + ]
C) Move the # you just wrote to the other side of the = sign
and change its sign (always makes it a negative). Collect.
1) (x )2 + c = y
– 2
D) Now it is in Vertex form: (x )2 + c = y

3. Completing the Square of x2 + bx + c
Examples: ½ the #, square the #. Move it on back.
x2 + 6x + 5 = y ) x2 – 12x + 4 = y
(x + 3) = y (x – 6) = (-6)2 + y
(x + 3) = 9 + y (x – 6) = 36 + y
– – 36
(x + 3)2 – 4 = y (x – 6)2 – 32 = y

4. Completing the Square of x2 + bx + c
IV.. Solve using Completing the square (finds solutions).
A) Complete the square ( see part II ), but change “y” to a 0.
1) DON’T move it on back.
B) Circle the (x – h)2 part (don’t include the “a” term in circle).
C) Isolate the circled part (Steps for Solving Equations).
D) Get rid of the exponent with a radical (sq root both sides).
E) Simplify the radical & solve for x (radicals give 2 answers).

5. Completing the Square of x2 + bx + c
IV.. Solve using Completing the square (finds solutions).
A) Complete the square ( see part II ), but change “y” to a 0.
1) DON’T move it on back.
B) Circle the (x – h)2 part (don’t include the “a” term in circle).
C) Isolate the circled part (Steps for Solving Equations).
D) Get rid of the exponent with a radical (sq root both sides).
E) Simplify the radical & solve for x (radicals give 2 answers).
Examples:
x2 + 6x + 5 = 0 6) x2 – 12x + 4 = 0
(x + 3)2 + 5 = step A (x – 6)2 + 4 = (–6)2
(x + 3)2 + 5 = step B (x – 6)2 + 4 = 36
–5 – step C –4 –4
√ (x + 3) = √ step D √ (x – 6)2 = √ 32
x + 3 = 2 or – step E x – 6 = 4√ 2 or – 4√ 2
– –
x = – 1 , – x = 6 ± 4√ 2