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 ax2+ 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
2x2 + 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 x2 unity (that is 1) by dividing the whole equation by the coefficient of x2.
2x2 + 3x – 2 = 0
From the equation above, the coefficient of x2 is 2; therefore, we divide through with 2.
(2x2/2) + (3x/2) – (2/2) = 0
X2 + 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.
X2 + 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.
X2 + 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, X2 + 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
X2 + 3x/2 + (¾)2 = 1 + (¾)2
Open brackets
X2 + 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
2x2 + 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 x2 unity (that is 1) by dividing the whole equation by the coefficient of x2.
2x2 + 8x –8 = 0
From the equation above, the coefficient of x2 is 2; therefore, we divide through with 2.
2x2/ 2 + 8x/2 – 8/2 = 0
X2 + 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.
X2 + 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.
X2 + 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, X2 + 2x = 4, the coefficient of x = 4
4 X ½ = 2
Add the square of 2 to both sides of the equation
X2 + 2x + (2)2 = 4 + (2)2
Open brackets
X2 + 2x + (2)2 = 4 + 4
X2 + 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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