Math Lesson 9.5.4 - Solving a Quadratic Equation by Completing the Square

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Welcome to our Math lesson on Solving a Quadratic Equation by Completing the Square, this is the fourth lesson of our suite of math lessons covering the topic of Quadratic Equations, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Solving a Quadratic Equation by Completing the Square

So far, we tried to solve a quadratic equation by trying to guess the coefficients and constant of the binomial expression. However, it was a bit challenging to find the right values, as we had to also take into account the rest of the expression, which must also fit the binomial expression. Another limitation of our reasoning consists of the fact that the coefficient a is not always a perfect square. Therefore, we must find a standard method for solving quadratic equations, that consists of completing the square. We must therefore try to express a given quadratic equation

in the form

where p and q are numbers.

The procedure applied to complete the square is as follows:

Step 1: First of all, we find the expression in the brackets. For this, we must express the squared part in the form

where b is the coefficient preceding x in the original equation. Thus, it is obvious that p = B/2.

Step 2: Expanding the last expression yields

Step 3: Check the difference between the old and new constant.

For example, in the equation x² - 4x + 3 = 0, we have a = 1, b = -4 and c = 3. Thus,

Given that the original equation is

we have q = -1 because

Therefore, the original quadratic equation is factorized by completing the square as

You may wonder why one needs to complete the square in quadratic equations. In fact, completing the square gives us the possibility to solve the quadratic equation by solving two first - order equations with one variable. In our example, we have

Thus, since √1 = -1 or +1, we obtain two different first - order equations with one variable:

and

The first equation gives x¹ = - 1 + 2 = 1, while the second one gives x² = 1 + 2 = 3.

Example 4

Solve the following quadratic equations by completing the square.

1. x² - 2x - 3 = 0
2. x² - 8x + 7 = 0

Solution 4

1. We have a = 1, b = -2 and c = -3. Thus, since B/2 = -1, we have for the part in brackets of the quadratic equation when trying to complete the square:
   (x + B/2)² = (x - 1)²
   = x² - 2x + 1
   The original equation was
   x² - 2x - 3 = 0
   Thus, we have
   x² - 2x - 3
   = (x² - 2x + 1) - 4
   = (x + B/2)² - 4
   = (x - 1)² - 4
   Therefore, this quadratic equation after completing the square is
   (x - 1)² - 4 = 0
   To solve this equation, we have to express it as
   (x - 1)² = 4
   Thus,
   x - 1 = √4
   = ± 2
   Hence,
   x₁ - 1 = - 2
   x₁ = - 2 + 1
   x₁ = - 1
   and
   x₂ - 1 = 2
   x₂ = 2 + 1
   x₂ = 3
2. We have a = 1, b = -8 and c = 7. Thus, since B/2 = -8/2 = -4, we have for the part in brackets of the quadratic equation when trying to complete the square:
   (x + B/2)² = (x - 4)²
   = x² - 8x + 16
   The original equation was
   x² - 8x + 7 = 0
   Thus, we have
   x² - 8x + 7 = 0
   (x² - 8x + 16) - 9 = 0
   (x + B/2)² - 9 = 0
   (x - 4)² - 9 = 0
   To solve this equation, we have to express it as
   (x - 4)² = 9
   Thus,
   x - 4 = √9
   = ± 3
   Hence,
   x₁ - 4 = - 3
   x₁ = - 3 + 4
   x₁ = 1
   and
   x₂ - 4 = 3
   x₂ = 3 + 4
   x₂ = 7

Remark! You can quickly check your work by substituting the roots found in the original equation. Thus, since the roots in the exercise 4a were - 1 and 3, we have

and

More Quadratic Equations Lessons and Learning Resources

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1. Completing The Square Feedback. Helps other - Leave a rating for this completing the square (see below)
2. Equations Math tutorial: Quadratic Equations. Read the Quadratic Equations math tutorial and build your math knowledge of Equations
3. Equations Video tutorial: Quadratic Equations. Watch or listen to the Quadratic Equations video tutorial, a useful way to help you revise when travelling to and from school/college
4. Equations Revision Notes: Quadratic Equations. Print the notes so you can revise the key points covered in the math tutorial for Quadratic Equations
5. Equations Practice Questions: Quadratic Equations. Test and improve your knowledge of Quadratic Equations with example questins and answers
6. Check your calculations for Equations questions with our excellent Equations calculators which contain full equations and calculations clearly displayed line by line. See the Equations Calculators by iCalculator™ below.
7. Continuing learning equations - read our next math tutorial: The Quadratic Formula

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