Mastering the Complete the Square Method: A Step-by-Step Guide to Solving Quadratic Equations with Ease

Steps to Complete the Square

Completing the square is a method used to solve quadratic equations or manipulate quadratic expressions

Completing the square is a method used to solve quadratic equations or manipulate quadratic expressions. It involves rearranging the quadratic expression into a perfect square trinomial. Here are the steps to complete the square:

Step 1: Start with a quadratic expression in the form ax^2 + bx + c, where a, b, and c are constants.

Step 2: If the coefficient of x^2 (a) is not 1, divide the entire equation by a so that the coefficient becomes 1.

Step 3: Move the constant term (c) to the right side of the equation.

Step 4: Take half of the coefficient of x (b/2) and square it [(b/2)^2]. This will give you the constant term that makes the trinomial a perfect square.

Step 5: Add the constant term you found in step 4 to both sides of the equation.

Step 6: On the left side of the equation, factor the quadratic expression into a perfect square trinomial. To do this, write the expression as (x + b/2)^2.

Step 7: Simplify the right side of the equation.

Step 8: Solve for x by taking the square root of both sides. Remember to consider both positive and negative square roots.

Step 9: Write your answer as x = m or x = n, where m and n are the simplified versions of the square root.

Let’s go through an example:

Example:
Solve the quadratic equation x^2 + 6x + 5 = 0 by completing the square.

Step 1: Start with the quadratic equation x^2 + 6x + 5 = 0.

Step 2: The coefficient of x^2 is 1, so we can move to the next step.

Step 3: Move the constant term 5 to the right side of the equation: x^2 + 6x = -5.

Step 4: Take half of the coefficient of x (6/2) and square it: (6/2)^2 = 9. This will be the constant term required to make the trinomial a perfect square.

Step 5: Add 9 to both sides of the equation: x^2 + 6x + 9 = -5 + 9.

Step 6: On the left side, factor the quadratic expression into a perfect square trinomial: (x + 3)^2 = 4.

Step 7: Simplify the right side: (x + 3)^2 = 4.

Step 8: Solve for x by taking the square root of both sides: x + 3 = ±√4.

Step 9: Simplify the square root of 4: x + 3 = ±2.

Now, let’s solve for x:

Case 1: x + 3 = 2
Subtract 3 from both sides: x = 2 – 3 = -1.

Case 2: x + 3 = -2
Subtract 3 from both sides: x = -2 – 3 = -5.

Therefore, the solutions to the quadratic equation x^2 + 6x + 5 = 0 are x = -1 and x = -5.

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