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Completing the Square

Completing the Square

If a given quadratic equation can be factored to a perfect square of a linear binomial, it can be solved easily by equating the resulting binomial to 0 and solving it. For example, if we factor a quadratic equation to yield

then we can proceed to the final solution as follows:

However, it's difficult to directly reduce many quadratic equations to a perfect square. For these quadratics, we use a method called completing the square.

Completing the square definition

Using the completing the square method, we try to obtain a perfect square trinomial on the left hand side of the equation. We then proceed to solve the equation using the square roots.

Using the completing the square method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equation.

Completing the square formula

In this article, we will go through the more formal steps of the completing the square method. But first, in this section, we look at a bit of a cheat sheet for solving quadratic equations by completing the square.

Given a quadratic equation of the form,

we convert it into

Directly implementing this formula will also give you the answer.

Completing the square method

While you can directly use the formula stated above, there is a more deliberate step-by-step method to solving quadratic equations using the completing the square method.

Note that in exams you would need to solve using the step-by-step method, so it is a good idea to get familiar with the process.

If you are given a quadratic equation of the form , follow the steps below to solve it using the completing the square method:

  1. If a (coefficient of x2) is not 1, divide each term by a.

    This yields an equation of the form

  2. Move the constant term () to the right hand side.

    This yields an equation of the form

  3. Add the appropriate term to complete the square of the left hand side of the equation. Do the same addition on the right hand side to keep the equation balanced.

    Hint: the appropriate term should be equal to .

    The equation should now be in the form

  4. Now that you have a perfect square on the left hand side, you can find the roots of the equation by taking square roots.

Let us take a look at some examples to illustrate this.

Completing the square examples

Here are a few examples with solutions for completing the squares.

Solve for x :

Solution:

Step 1 – Divide each term by 2.

Step 2 –Move the constant term to the right-hand side.

Step 3 –Complete the square by adding 4 to both sides.

Step 4 –Find the roots by taking square roots.

Thus, the roots of the equation are

Solve for x :

Solution:

Step 1 – The coefficient of x2 is 1. So we can move on to step 2.

Step 2 – Move the constant term to the right-hand side.

Step 3 – Complete the square by adding 9 to both sides.

Step 4 – Find the roots by taking square roots.

Thus, the roots of the equation are

Remember the formula we discussed earlier in the article. Let us now try solving the above example directly using the formula. Do keep in mind that during the exam, you should use the method described above instead of directly inserting values into the formula.

Solve for x :

Solution:

Let us directly put the equation in the form

This give us

which is exactly what we got using the method in the previous example. From here on, you can follow the process in the same way as in the above example to obtain the roots, 7 and -1.

While you should not solve questions like this in a written examination, this can be a very useful short cut if you need to rapidly find the roots of a quadratic equation or if you want to cross-check whether the answer you have found using the former method is accurate.

Completing the Square - Key takeaways

  • Many quadratic equations are very difficult to directly reduce to a perfect square. For such quadratics, we can use the method called completing the square.
  • Using the completing the square method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equation.
  • Using the completing the square method we transform a quadratic equation of the forminto

Frequently Asked Questions about Completing the Square

Using the completing the square method, we add or subtract terms to both sides of a quadratic equation until we have a perfect square trinomial on one side of the equation.

Using the completing the square method we transform a quadratic equation of the form ax²+bx+c=0 into (x+d)²=e, where d=b/2a and e=b²/4a² - c/a

If you are given a quadratic equation of the form ax²+bx+c=0, follow the steps below to solve it using the completing the square method:


  1. If a (coefficient of x2) is not 1, divide each term by a. 
  2. Move the constant term to the right hand side.
  3. Add the appropriate term to complete the square of the left hand side of the equation. Do the same addition on the right hand side to keep the equation balanced.
  4. Now that you have a perfect square on the left hand side, you can find the roots of the equation by taking square roots. 

Beolow is an example of completing the squares:

Solve for x : Solution

Step 1 – Divide each term by 2.


Step 2 –Move the constant term to the right-hand side.


Step 3 –Complete the square by adding 4 to both sides.


Step 4 –Find the roots by taking square roots.


Thus, the roots of the equation are


Final Completing the Square Quiz

Question

Solve for x:

x²+2x+1=0

Show answer

Answer

x=-1

Show question

Question

What is a quadratic equation?

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Answer

A quadratic equation is an equation that can be expressed in the form ax²+bx+c=0, where a≠0.

Show question

Question

Solve for x: x²+x-6=0

Show answer

Answer

x=-3, x=2

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Question

Find the roots of 

2x²+4x=0

Show answer

Answer

0, -2

Show question

Question

What is the completing the square method?

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Answer

Using the completing the square method, we add or subtract terms to both sides of the equation until we have a perfect square trinomial on one side of the equation.

Show question

Question

Find the roots of 

x²-6x-7=0

Show answer

Answer

7, -1

Show question

Question

Find the roots of 

x²+x-6=0


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Answer

2, -3

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Question

Find the roots of 

x²+8x+15=0

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Answer

-3, -5

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Question

Find the roots of 

4x²+15x+9=0

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Answer

-3, -0.75

Show question

Question

Find the roots of 

2x²-6x-8=0

Show answer

Answer

4,-1

Show question

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