Solving Quadratic Equations by Completing the Square: Everything You Need to Know!

Solving Quadratic Equations by Completing the Square
Solving Quadratic Equations by Completing the Square

Quadratic equations are an important concept in Algebra, but they can be intimidating to some high school students, particularly those who are not well-versed in math.

I find that one of the most useful and effective methods for solving quadratic equations is called completing the square. By understanding how to complete the square, any student can easily solve any quadratic equation.

Read on to explore a step-by-step process of solving quadratic equations by completing the square. If you have been wondering why mathematicians use the letter X to represent the unknown in maths, I wrote a whole article that I invite you to read.

What Is Completing The Square Method?

Along with the quadratic formula, completing the square method is one of the most effective techniques for finding the roots of the given quadratic equation. First, you will need to convert the given equation into a perfect square when using the completing the square method. 

How To Solve Quadratic Equations By Completing The Square?

Completing the square is particularly useful when you are working with quadratic equations in the standard form of ax² + bx + c = 0.

To solve a quadratic equation using completing the square, you need to follow four simple steps:

Step 1: To start the process, you will need to move the constant term (c) to the other side of the equation, so you’re left with ax² + bx = -c. In some cases, you will need to rewrite the equation so that the x² term has a coefficient of 1.

  • For example, the equation 2x² + 12x = 10 would be rewritten as x² + 6x = 5.

Step 2: Take half of the coefficient of x, and square it. Then, add this value to both sides of the equation. This step allows you to create a perfect square trinomial, which can be factored into a binomial squared.

  • For example, if the equation is x² + 6x = 5, the coefficient of x is 6. Half of 6 is 3, and 3² is 9. Therefore, we add 9 to both sides, resulting in the equation x² + 6x + 9 = 14.

Step 3: Factor the left side of the equation into a perfect square. In this example, the left side factors to (x + 3)². Therefore, our equation is now (x + 3)² = 14.

Step 4: Solve for x by taking the square root of both sides of the equation. However, it is important to take the positive and negative square roots, as both values can be valid.

  • In this example, we have (x + 3) = ±√14. By subtracting 3 from both sides, we get x = -3 ± √14.

Example Of Completing The Square Method

Let’s look at another example of completing the square in action.

  • Say we have the equation 2x² + 8x – 6 = 0.
  • To start, we will move the constant term to the other side of the equation, so we have 2x² + 8x = 6.
  • Then, we will take half of the coefficient of x (which is 4) and square it (16).
  • We will add 16 to both sides of the equation, giving us 2x² + 8x + 16 = 22.
  • Now we have a perfect square trinomial and can factor it into (x + 2)² = 11.
  • Taking the square root of both sides, we get x + 2 = ±√11. Solving for x, we get x = -2 ± √11.

Alternative Way To Completing The Square

Another use for completing the square is finding the maximum or minimum point of a quadratic equation, which is also known as the vertex of the parabola created by the equation.

To find the vertex, begin by putting the equation in standard form. Then, use the same steps as completing the square to find the vertex.

  • Set x equal to -b/2a, which will give you the x-value of the vertex.
  • And then, substitute the x-value of the vertex into the original equation to find the y-value of the vertex.

When Should You Use The Completing The Square Method?

Completing the square is a method used to find the minimum or maximum values of quadratic functions. In addition, you can also use the completing the square method to change or simplify the form of algebraic expressions. The Completing the square technique is also useful when solving quadratic equations.

If you want to learn more about completing the square, I encourage you to check Khan Academy and watch this video or the video below.

What to read next:

Wrapping Up

Completing the square is an easy and efficient method for solving quadratic equations. It can be used in a variety of situations, from finding the roots of an equation to locating the vertex of a parabola.

By following the four steps outlined in this article, you can solve any quadratic equation you encounter.

Remember, practice makes perfect, and the more you practice solving quadratic equations by completing the square, the more confident you’ll become.

Altiné

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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