How to Complete the Square and Solve Quadratic Equations | A Step-by-Step Guide

Steps to Complete the Square

To complete the square for a quadratic equation, follow these steps:

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To complete the square for a quadratic equation, follow these steps:

1. Start with a quadratic equation in the form of ax^2 + bx + c = 0, where a, b, and c are constants.

2. If the coefficient of x^2, ‘a’, is not equal to 1, divide the entire equation by ‘a’ to make it equal to 1. This step is necessary as completing the square works best when the coefficient of x^2 is 1.

3. Move the constant term ‘c’ to the right side of the equation, so that you have a binomial on the left side containing x terms. The equation then becomes ax^2 + bx = -c.

4. Take half of the coefficient of x, ‘b’, and square it. Add this value to both sides of the equation. This step is crucial to completing the square. The equation now becomes ax^2 + bx + (b/2)^2 = (b/2)^2 – c.

5. Factor the left side of the equation. It should be a perfect square trinomial. The equation is now (x + b/2)^2 = (b/2)^2 – c.

6. Take the square root of both sides of the equation. Keep in mind that there may be two solutions, considering the positive and negative square roots. The equation now becomes x + b/2 = ± √((b/2)^2 – c).

7. Finally, isolate the variable x by subtracting b/2 from both sides of the equation. The final solutions are x = -b/2 ± √((b/2)^2 – c).

These steps allow you to complete the square for any quadratic equation and find the solutions to it.

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