Download Completing the Square Worksheet

CCGPS Geometry - “Completing the Square” Worksheet
Name ____________________
“Completing the Square” is a technique (like factoring or quadratic formula) for solving quadratic
equations from standard form. “Completing the square” will ALWAYS work. Sometime the numbers can
be ugly (fractions) but it will find the answers if they are rational (real), irrational (real) or imaginary. If
the solutions are irrational or real, they will be conjugates. In a sense, “completing the square” will
mathmagically turn a quadratic with x2 and x into just a single “x”. Once this is done, you can
| | so
“unwrap”/solve by isolating the variable. Just remember ± when square rooting both sides. √
you must consider both + or - . Here is a general example and some examples worked out.
This is the original equation.
Move the constant over to the other side.
Take half of the x-term (that is, divide it by two) (and don't
forget the sign!), and square it. Add this square to both sides of
the equation. (Addition Prop of Equality)
Convert (Factor) the left-hand side to squared form. Simplify
the right-hand side.
( )
(
)
( )
( )
Square-root both sides. Remember to do "±" on the right-hand
side. Simplify.
√
( )
√
( )
Solve for "x =". Remember that the "±" gives you two solutions.
Simplify as necessary.
Completing the square with Leading Coefficient of 1 (ex. 1x2)
This is the original equation.
Move the constant over to
the other side.
Take half of the x-term
(that is, divide it by two)
(and don't forget the sign!),
and square it. Add this
square to both sides of the
equation. (Addition Prop of
Equality)
x 2 + 6x – 7 = 0
x2 - 4x + 1 = 0
x2 - 3x + 1 = 0
x2 - 8x + 24 = 0
x 2 + 6x
x2 - 4x
x2 - 3x
x2 - 8x
x2 + 6x +9
=7
= 7 +9
x2 - 4x +4
= -1
= -1 +4
Convert (Factor) the lefthand side to squared
form. Simplify the righthand side.
(x + 3)2 = 16
(x - 2)2 = 3
Square-root both sides.
Remember to do "±" on the
right-hand side. Simplify.
x+3=±4
x-2=±√
Solve for "x =". Remember
that the "±" gives you two
solutions. Simplify as
necessary.
x=–3±4
= – 3 – 4, –3 + 4
x = –7, +1
x-=2±√
OR
x-=2+√ ,2-√
= -1
x2 - 3x +
(
= -1 +
x2 - 8x +16 = -24+ 16
=
(x - 4)2 = -8
= ±√
x - 4 =± √
x - 4 =± √
)
=±
x =
= -24
±
√
√
x =4±
√
For help on #7-12 goto http://www.themathpage.com/alg/complete-the-square.htm
1. x2 – 4x + 3 = 0
2. x2 + 2x – 3 = 0
3. x2 – x = 42
4. x2 – 20x = -96
5. x2 + 5x = 66
6. x2 – 7x + 9 = 0
7. x2 - 2x – 2 = 0
8. x2 – 10x +20 = 0
9. x2 - 4x + 13 = 0
10. x2 + 6x + 29 = 0
11. x2 – 5x – 5 = 0
12. x2 + 3x + 1 = 0
13. x2 = 14x – 72
14. 16x + 60 = -x2
15. 15x = x2 + 4