# Quadratic Equation Using Completing The Square Method

This article will be a useful guide to help you understand how to solve quadratic equation using completing the square method. It will show you a step-by-step means for solving simple and complex quadratic equation using completing the square method. Ensure to read to the end.

**What Is Quadratic Equation?**

In our previous article on Quadratic equations using Factorization method, we explained everything about the concept of quadratic equation. As a reminder, a quadratic equation is an equation which takes the general form of ax^{2}+ bx + c = 0 where the highest power of the unknown is 2.

We highlighted that there were several methods of solving quadratic equation which includes:

- Factorization method
- Completing the square method
- Formula method and
- Graphical method

In this article, we shall concentrate on solving quadratic equation using completing the square method.

**Quadratic Equation Using Completing The Square Method**

Factorization method cannot solve all quadratic equations as it has a limited range of application. However, completing the square method has a wide range of application and can be referred to as a general method. It can be used to solve quadratic equations which cannot be solved by the factorization method.

Solving a quadratic equation using completing the square method involves making the expression of the equation a perfect square which can then be factorized to get your answers. We are going to evaluate some of the ways to apply this method using different examples below.

**Example 1**

2x^{2 }+ 3x – 2 = 0. Solve the quadratic equation using completing the square method

**Solution**

**Step 1: **Write out the given equation and proceed to making the coefficient of x^{2 }unity (that is 1) by dividing the whole equation by the coefficient of x^{2}.

2x^{2 }+ 3x – 2 = 0

From the equation above, the coefficient of x^{2} is 2; therefore, we divide through with 2.

(2x^{2}/2) + (3x/2) – (2/2) = 0

X^{2} + 3x/2 – 1 = 0

**Step 2: **Ensure that the right hand side of the equation is 0. If it is not 0, then, rearrange the equation to make it 0.

X^{2} + 3x/2 – 1 = 0

**Step 3: **Transfer the constant term (the term without an unknown) to the right hand side of the equation (in our case, it is -1). Ensure to take note of the sign when moving it across the equality sign.

X^{2} + 3x/2 = 1

**Step 4: **Make the Left Hand Side (LHS) of the equation a **Perfect Square. **To do this, you have to add the square of half of the coefficient of x to both sides of the equation

From our equation, X^{2} + 3x/2 = 1, the coefficient of x = 3/2

3/2 X ½ = ¾

Add the square of ¾ to both sides of the equation

X^{2} + 3x/2 + (¾)^{2} = 1 + (¾)^{2}

**Open brackets**

X^{2} + 3x/2 + 9/16 = 1 + 9/16

1 + 9/16 = 25/16

**Step 5: **You then have to factorize the LHS.

(x + ¾)^{2 }= 25/16

**Step 6: **You have to take the square root of both sides of the equation

√(x + ¾)^{2 }= √ 25/16

x + ¾ = 5/4

x = -¾ ± 5/4

= (-3+5)/4 or (-3-5)/4

x = 2/4 or -8/4

x = ½ or -2

From the above example, you can see that solving quadratic equation using completing the square method can be a straightforward process if you follow the steps accordingly. We are going to solve one more example.

**See Also:**

**Quadratic Equation: How To Solve Using Factorization Method**

###### Simultaneous Equation- Meaning, Methods and Examples

**Example 2: Quadratic Equation Using Completing The Square Method**

2x^{2 }+ 8x –8 = 0. Solve the quadratic equation using completing the square method

**Solution**

**Step 1: **Just like for the previous example, write out the given equation and proceed to making the coefficient of x^{2 }unity (that is 1) by dividing the whole equation by the coefficient of x^{2}.

2x^{2 }+ 8x –8 = 0

From the equation above, the coefficient of x^{2} is 2; therefore, we divide through with 2.

2x^{2}/ 2^{ }+ 8x/2 – 8/2 = 0

X^{2} + 4x – 4 = 0

**Step 2: **You have to ensure that the right hand side of the equation is 0. Rearrange the equation to make it 0. In this equation, the RHS is 0.

X^{2} + 4x – 4 = 0

**Step 3: **Transfer the constant term (the term without an unknown) to the right hand side of the equation (in our case, it is -4). As usual, you have to ensure that you take note of the negative sign when moving it across the equality sign.

X^{2} + 4x = 4

**Step 4: **Make the Left Hand Side (LHS) of the equation a **Perfect Square. **As already explained, you have to add the square of half of the coefficient of x to both sides of the equation

From our equation, X^{2} + 2x = 4, the coefficient of x = 4

4 X ½ = 2

Add the square of 2 to both sides of the equation

X^{2} + 2x + (2)^{2} = 4 + (2)^{2}

**Open brackets**

X^{2} + 2x + (2)^{2} = 4 + 4

X^{2} + 2x + (2)^{2} = 8

**Step 5: **You then have to factorize the LHS. This is done easily by taking the unknown and the number that has a square sign.

(x + 2)^{2 }= 8

**Step 6: **You have to take the square root of both sides of the equation

√(x + 2)^{2 }= √ 8

x + 2 = √8

x = -2 ± √8

Then, x = -2 + √8 or-2 – √8

x = -2 + 2.8 or -2-2.8

Therefore, x = 0.8 or -4.8

These two examples have clearly shown you how to solve quadratic equation using completing the square method. You can now go ahead to solve more exercises on your own.

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