Quadratic Equation Using Completing The Square Method
This article will be a helpful guide to help you understand how to solve quadratic equations using completing the square method. It will show you a step-by-step means for solving simple and complex quadratic equations using completing the square method. Ensure to read to the end.
What Is Quadratic Equation?
In our previous article on Quadratic equations using the Factorization method, we explained everything about the quadratic equation. As a reminder, a quadratic equation is an equation that 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 equations, which include:
- Factorization method
- Completing the square method
- Formula method and
- Graphical method
This article will solve quadratic equations by completing the square method.
Quadratic Equation Using Completing The Square Method
The factorization method cannot solve all quadratic equations as it has limited applications. However, completing the square method has many applications and can be referred to as a general method. It can be used to solve quadratic equations that the factorisation method cannot solve.
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 make 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, 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 equation’s Left Hand Side (LHS) 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 equations using the completing the square method can be straightforward 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: 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 must ensure that you note the negative sign when moving it across the equality sign.
X2 + 4x = 4
Step 4: Make the equation’s Left Hand Side (LHS) 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 quickly by taking the unknown and the number with 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 clearly show you how to solve quadratic equations using the completing the square method. You can now go ahead and solve more exercises on your own.
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