## Completing the Square

### Completing the square is a technique used in algebra to manipulate a quadratic equation into a perfect square trinomial

Completing the square is a technique used in algebra to manipulate a quadratic equation into a perfect square trinomial. This method is particularly useful when solving quadratic equations, graphing quadratic functions, or simplifying expressions involving quadratic terms.

To complete the square for a quadratic equation in the form of ax^2 + bx + c = 0, follow these steps:

1. Ensure that the coefficient of x^2 (the “a” value) is equal to 1. If it is not, divide the entire equation by a to make it equal to 1. For example, if we have 3x^2 + 6x + 2 = 0, we would divide through by 3 to get x^2 + 2x + 2/3 = 0.

2. Move the constant term (the “c” value) to the right side of the equation, isolating the x^2 and x terms. With the previous example, we would rewrite the equation as x^2 + 2x = -2/3.

3. Take half of the coefficient of the x term (the “b” value), square it, and add it to both sides of the equation. This step completes the square on the left side of the equation. For our example, the coefficient of x is 2, so we take half of 2, which is 1, square it to get 1, and add it to both sides:

x^2 + 2x + 1 = -2/3 + 1.

4. Rewrite the left side of the equation as a perfect square trinomial. The left side of the equation should now be in the form of (x + p)^2, where “p” represents the number we added to both sides. In our example, the left side becomes (x + 1)^2, while the right side simplifies to 1/3.

5. Take the square root of both sides of the equation and solve for x. Remember to consider both the positive and negative square roots. In our example, the square root of (x + 1)^2 is x + 1, so we have:

x + 1 = ±√(1/3).

6. Finally, simplify the equation to get the solutions for x by subtracting 1 from both sides:

x = -1 ± √(1/3).

These are the solutions to the quadratic equation after completing the square. Completing the square helps rewrite a quadratic equation in a more useful form, making it easier to solve or graph.

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