Quadratic Formula

Tiling the Square: Group Activity

Here’s a new look at a different approach to solving quadratics . In the 9^th Century C. E., a
Babylonian mathematician named Al-Khwarizmi was the principle creator of the process that is now
known as completing the square. We’ll use algebra tiles to understand how geometry can be used to
solve algebraic equations.

Consider the quadratic equation x² + 6x – 4 = 12.
1. Write the equation in ax ² + bx = -c form.

2. Arrange the 6”x“ tiles around the 1 “x²” tile to form part of a square. What did you have do to
the 6 “x” tiles in order to make the partial square?
Draw your results .

3.Now that you have a partial square, how many “1” tiles can you fit to complete the square?
 _____________

4. What are the dimensions of our completed square?
 __________________ and _______________ or ________________

5. So, now rewrite the left hand side of your equation by substituting your completed square for
the perfect square trinomial you have created in step 3.

6. Solve for x.

Work in groups to solve the fol lowing equations by completing the square.
a. Write the equations so that the coefficient of the x² term is 1.
b. Make sure that your equation is in ax² + bx = -c form.
c. Use the correct amount of algebra tiles to arrange tiles into a partial square.
d. Complete the square with “1” tiles and rewrite the equation.
e. Solve for x.

7. 4x² + 32x = 132

a.
b.
c.
d.
e.

8. 2x² + 20x -6 = 16

a.
b.
c.
d.
e.

Extension
Consider the equations on your group worksheet as ax² + bx = -c

We wanted to make sure that a = 1, so we divided through by a to get .

Now, when we look at b in Problem 9, .

What relationship do we see that corresponds to the sides of our completed square? In other
words, if (x + d) was the side of our square, what is the relationship between and d?
(Hint: d = ?).

Notice that is the number of “1” tiles we added to complete the square each time.

****Also, do not forget that we added this to both sides.
So, could we use this to complete the square without algebra tiles? Yes.

If we add to both sides, what does the equation become?



At this point we rewrote the equation to get:


Or


If we simplify the right hand side of the equation, we now have:

Then we took the square root of both sides to get:



Now if we solve for x, the equation becomes



This formula is what is known as the Quadratic Formula.
***Remember, to use the Quadratic Formula, the quadratic equation you are solving must be in
standard form (ax² + bx + c = 0).

Now we have three ways of solving quadratic equations in the form: ax² + bx –c = 0.
I.

II.

III.

Answer Key.

Falc oner Football

Area = 8x² + 8x +2

Let x = 29.5 yards, then Area = 7200 yards²
Grid Multiplication with Tiles
1.

2.________6_______Number of x² tiles. Now write the product below:
3.________8_______Number of x tiles.
4.________2_______Number of 1 tiles. 5. 6x² + 8x + 2________________

6. (2x – 4) and (-x – 2)

7.

8. ________-2_______Number of x² tiles. Now write the product below:
9. ________0 _______Number of x tiles.
10.________8_______Number of 1 tiles. 11. -2x² + 8________________
12. Factors: ____-3x - 1 __ and ___-2x + 2_______.

13.

14. Final product: ___6x² - 4x -2___________________

Multiply Trinomials!!!

Tiling the Square Activity

Split them in half.

Extension


I. Factoring, if possible.
II. Completing the square by rewriting the equation and solving for x.
III. Using the Quadratic Formula.