Mastering Completing the Square | The Ultimate Guide to Transform Quadratic Equations and Unlock Their Secrets

Steps to Complete the Square

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

Completing the square is a technique used in algebra to transform a quadratic equation into a perfect square trinomial. This process is helpful in solving quadratic equations, graphing quadratic functions, and finding the vertex of a parabola. Here are the steps to complete the square:

Step 1: Write the quadratic equation in the standard form
A quadratic equation is typically written in the standard form: ax^2 + bx + c = 0, where a, b, and c are constants, and a ≠ 0.

Step 2: Divide the entire equation by the coefficient of x^2
Divide both sides of the equation by the coefficient (a) of the x^2 term to make the coefficient of x^2 equal to 1. This step ensures that the equation is in the standard form.

Step 3: Move the constant term (c/a) to the other side of the equation
Subtract the constant term (c/a) from both sides of the equation, moving it to the side without x terms.

Step 4: Create a perfect square trinomial
On the side with the x terms, add (b/2a)^2, which is the square of half the coefficient of the x term, to both sides of the equation. This step completes the square on the left side of the equation.

Step 5: Factor the left side of the equation
Factor the left side of the equation into a perfect square trinomial. This expression will be of the form (x + k)^2, where k is a constant.

Step 6: Simplify the right side of the equation
Simplify the right side of the equation by combining like terms if possible. This step is usually straightforward since the purpose of completing the square is to create a perfect square trinomial on the left side.

Step 7: Solve for x
Take the square root of both sides of the equation to solve for x. Remember to consider both the positive and negative square root.

Step 8: Write the solutions
Write the solutions for x in the form x = k or x = j, where k and j are constants.

By following these steps, you can complete the square for any quadratic equation and find its solutions or rewrite it in vertex form. Remember to check your solutions by substituting them back into the original equation to ensure they are valid.

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