See Completing the Square for a discussion of the process.

Solving quadratic equations by completing the square is often overpowered by an "offspring" of this process, namely, the quadratic formula. Once the quadratic formula was derived, it was no longer necessary to use the process of completing the square to solve "each" quadratic equation. Even though completing the square is often overlooked in favor of the quadratic formula, it is still a valuable skill that is needed in other mathematical situations. So, don't overlook the power of this skill.

statement
Creating a perfect square trinomial on the left side of a quadratic equation, after moving the constant (number) to the right, is the basis of a process called completing the square.

expin1 Find the solutions for:    x2 = 3x + 18
ex1left
Solutions: x = 6, x = 3
(The leading coefficient is one.) Get the x-related terms on the left side. Keep the constant term on the right side.

Prepare the equation to receive the added value (boxes).
Take half of the x-term's coefficient and square it. Add this value to both sides (fill the boxes).

Combine terms on the right.


Factor the perfect square trinomial on the left side.


Take the square root of both sides. Be sure to consider "plus and minus", as we need two answers.

Solve for x. Clearly indicate your answers.

Prepare a check of the answers.
62 - 3(6) = 18 check
(-3)2 - 3(-3) = 18 check



expin1 Find the x-intercepts for:  4x2 - 8x - 32 = 0
csex2

x-intercepts are (4,0) and (-2,0)
The leading coefficient is NOT 1.

Divide all terms by 4 (the leading coefficient).

Move the constant to the right hand side.

Prepare the equation to receive the added value (boxes).
Take half of the x-term's coefficient and square it. Add this value to both sides (fill the boxes). Combine like terms.

Factor the perfect square trinomial on the left side.

Take the square root of both sides. Be sure to consider "plus and minus".

Solve for x. Clearly indicate your answers.

Prepare a check of the answers.
4(4)2 - 8(4) - 32 = 0 check
4(-2)2 - 8(-2) - 32 = 0 check


expin1 Find the solutions for:    x2 = 4x -1

complsq3

complsol3
Get the x-related terms on the same side (move 4x).
The leading coefficient is 1.

Prepare the equation to receive the added value (boxes).
Take half of the x-term's coefficient and square it. Add this value to both sides (fill the boxes). Combine like terms.

Factor the perfect square trinomial on the left side.

Take the square root of both sides. Be sure to consider "plus and minus".

Solve for x. Clearly indicate your answers.

Prepare a check of the answers.
check3n


expin1 Find the solutions for:  4x2 + 8x - 9 = 0
The leading coefficient is NOT 1.

Divide all terms by 4 (the leading coefficient).
[ Note: In some problems, this division process may create fractions, which is OK. Just be careful when working with the fractions.]

Move the constant to the right hand side.

Prepare the equation to receive the added value (boxes).
Take half of the x-term's coefficient and square it. Add this value to both sides (fill the boxes). Combine like terms.

Factor the perfect square trinomial on the left side.

Take the square root of both sides. Be sure to consider "plus and minus".

Solve for x. Clearly indicate your answers.

Prepare a check of the answers.



expin1 Find the solutions for:    x2 - 5x + 7 = 0

complex4

complsol4

Notice that this example involves the imaginary "i", and has complex roots of the form a + bi.

Answers written in a + bi form:

Read more about imaginary values.

These answers are not "real number" solutions. They do not have a place on the x-axis.
(The leading coefficient is one.) Move the constant to the right hand side.

Prepare the equation to receive the added value (boxes).
Take half of the x-term's coefficient and square it. Add this value to both sides (fill the boxes).

Get a common denominator on the right.

Factor the perfect square trinomial on the left side. Combine terms on the right. bewaresmall At this point, you have a squared value on the left, equal to a negative number. We know that it is not possible for a "real" number to be squared and equal a negative number.
____________________________________________

This problem involves "imaginary" numbers.

Take the square root of both sides. Be sure to consider "plus and minus". Notice the negative under the radical.

Solve for x. remember

Prepare a check of the answers.
check3

 

bullet Completing the Square with Algebra Tiles
Starting with x2 + 6x - 16 = 0, we rearrange x2 + 6x = 16 and attempt to complete the square on the left-hand side.
KEY: algebratileskey        See more about Algebra Tiles.
x2 + 6x = 16
Arrange the x2-tile and 6x-tiles to start forming a square. csat1
Notice how many 1-tiles are needed to complete the square.

csat2
Read the sides of the completed square.
csat3(x + 3)2 = 16 + 9

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