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# Completing the Square

Any quadratic equation can be solved using a technique called completing the square.

To use this method, we will rewrite the quadratic equation so that one side is a perfect square. Then we solve using the square root property.

Before we solve a quadratic equation in this way, letâ€™s learn how to complete the square.

To â€œcomplete the squareâ€ means to transform a binomial of the form x2 + bx into a perfect square trinomial by adding a constant term.

For example, letâ€™s complete the square for x2 + 6x.

We will use rectangles, called algebra tiles, to visualize the process.

â€¢ A square tile measuring x units on a side has area x2. We will use this tile to represent x2, the first term of x2 + 6x.

â€¢ A rectangular tile that is x units tall and 1 unit wide has an area of 1x. Since 6 Â· 1x is 6x, we will use six of these tiles to represent the second term of x2 + 6x.

Placed side by side, the tiles form a rectangle that represents x2 + 6x.

Now, we try to rearrange the tiles to form a square. To do this, we move one-half of the tall thin tiles. However, the result is not a complete square because the lower right portion is missing.

The missing piece is a 3-by-3 square.

Thus, to â€œcomplete the square,â€ we must add 3 Â· 3 = 9 unit tiles.

With the 9 new tiles, the area of the entire square is (x + 3)(x + 3) = (x + 3)2 = x2 + 6x + 9.

By adding 9 to x2 + 6x, we have created the perfect square (x + 3)2.

 Letâ€™s review the process we used to complete the square. We moved one-half of the x-tiles. x2 + 6x + ? Multiply the coefficient of x by Then we filled in the remaining space with a square of unit tiles. Square the result. 32 = 9 We added nine tiles to complete the square. x2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)2

This process holds in general:

To find the number needed to complete the square, multiply the coefficient of the x-term by , and then square the result.

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