Completing the Square

What is the Completing the Square Method?

Solving a quadratic equation can be simple if it can be factored into a perfect square of a linear binomial and set equal to 0. However, not all quadratic equations can be easily factored in this way. For these more complex equations, a method called completing the square is used.

The Definition of Completing the Square

The completing the square method involves manipulating an equation to create a perfect square trinomial on one side of the equation. This allows us to use the square root to find the solution.

The Completing the Square Formula

By rewriting a quadratic equation in the form x² + bx + c = 0, we can use the completing the square method to solve it. Our goal is to transform it into (x + p)² = q, where p and q are constants. With this formula, we can easily determine the roots of the equation.

The Step-by-Step Approach to Completing the Square

While the formula can be applied directly, it is important to understand the step-by-step process for solving quadratic equations using the completing the square method, especially during exams. Here is a guide on how to use this method:

• If the coefficient of x² is not 1, divide each term by that coefficient.
• Move the constant term to the right-hand side of the equation.
• Add the appropriate term to complete the square on the left-hand side of the equation. Do the same on the right-hand side to maintain balance.
• Take the square root of both sides to find the roots of the equation.

Examples of Completing the Square

Let's walk through a few examples of completing the square.

Example 1: Solve for x: x² + 8x + 16 = 0

Solution:

• Divide each term by 2: (x² + 8x + 16)/2 = 0/2
• Move the constant term to the right-hand side: x² + 8x = -16
• Add 4 to complete the square: x² + 8x + 4 = -16 + 4
• Take the square root of both sides: (x + 4)² = -12
• Find the values of x: x = -4 ± √(-12)
• Thus, the roots of the equation are x = -4 + 2√3 and x = -4 - 2√3

Example 2: Solve for x: x² + 6x = -9

Solution:

• Move the constant term to the right-hand side: x² + 6x + 9 = 0
• Add 9 to complete the square: x² + 6x + 9 = 0 + 9
• Take the square root of both sides: (x + 3)² = 9
• Find the values of x: x = -3 ± √9
• Thus, the roots of the equation are x = 0 and x = -6

Remember, while the formula can be directly used for solving quadratic equations, it is important to familiarize yourself with the step-by-step method for exams.

Bonus Shortcut: The formula we discussed earlier can be used directly to solve equations, but this should not be done during exams. Let's try it on our second example above.

Solution: Let's convert the equation to the form x² + bx = c. We get x² + 6x = 9. Now, we can use the formula x = (-b ± √(b² - 4ac))/2.

Plugging in our values, we get x = (-6 ± √(36 - 36))/2, which simplifies to x = -3 ± 0. Thus, we get the same roots as we did using the step-by-step method: x = 0 and x = -6.

This shortcut can be handy when you need to quickly find the roots of a quadratic equation, but it is important to use the step-by-step method in exams to ensure accuracy.

Key Takeaways

The completing the square method is a helpful approach for solving quadratic equations that cannot be easily factored. By manipulating the equation and obtaining a perfect square trinomial, we can easily find the roots of the equation. It is important to understand the step-by-step method for exams, and while the formula can be used directly, it is not recommended for exams. Use this method to efficiently solve quadratic equations and check your solutions for accuracy.

A Step-by-Step Guide to Using the Completing the Square Method

When faced with a quadratic equation in the form of ax²+bx+c=0, it can feel overwhelming. However, there is a method called completing the square that can help you solve these equations with ease.

Transforming Quadratic Equations Using the Completing the Square Method

Do you find yourself struggling with solving complicated quadratic equations in their original form? Don't worry, there is a method that can make them more manageable. By using the completing the square method, you can transform a quadratic equation into a simpler and more easily solvable form: (x+d)²=e. Here's a step-by-step guide:

Step 1: If the coefficient of x² (a) is not 1, divide each term by a to simplify the equation.

Step 2: Move the constant term (c) to the right-hand side of the equation. This will leave the left-hand side with only the coefficients of x² (a) and x (b).

Step 3: Complete the square by adding the appropriate term to both sides of the equation. Take the coefficient of x (b), divide it by 2, and square the result. This will give you the value of d, which will be used in the next step.

Step 4: Add d² to both sides of the equation. This will create a perfect square on the left-hand side, while the right-hand side will have an additional term of b²/4a².

Step 5: Take the square root of both sides of the equation. This will result in a simpler equation, where x is isolated on one side and the roots can be easily found on the other.

Now, let's see how this method works with an example:

Example: Solve for x: 2x²+4x+3=0

Solution:

Step 1: Divide each term by 2 to simplify the equation: x²+2x+1.5=0

Step 2: Move the constant term (1.5) to the right-hand side: x²+2x=-1.5

Step 3: Completing the square, add (2/2)²=1 to both sides: x²+2x+1=1-1.5

Step 4: Take the square root of both sides: (x+1)²=0.5

Step 5: Find the roots by taking the square root of both sides: x+1=±√(0.5)

Therefore, the roots of the equation are x=-1±√(0.5).