Solving Quadratic Equations by Completing the Square

Are you preparing for O-level exams?

Here is a quick crash course on solving Quadratic Equations by completing the square.

Before you read on, you might want to download this entire revision notes in PDF format to print it out, or to read it later.
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Solving Quadratic Equations by Completing the Square

 

To solve the quadratic equation ax2 + bx + c = 0 by completing the square, you can follow the steps below:

Step 1: Change coefficient of x2 equal to 1

a( x2 + bx/a + c/a) = 0

Step 2: Leave x2 and x terms on the Left Hand Side (LHS)

x2 + bx/a = – c/a

Step 3: Coefficient of x ÷2, square it, add to both sides

x2 + bx/a + (b/2a)2 = – c/a + (b/2a)2

Step 4: Factorise the LHS of the equation

(x + b/2a)2 = – c/a + (b/2a)2

Step 5: Simplify to solve for x

(x + b/2a)2 = – c/a + (b/2a)2

= –4ac/4a2 + b2/4a2

= (b2 – 4ac)/4a2

x + b/2a = ±√( (b2 – 4ac)/4a2 )

x = – b/2a  ±√( (b2 – 4ac)/4a2 )

x =( – b ±√( (b2 – 4ac) ) /2a

 

Does the answer look familiar to you?

Yes, it is the Quadratic Formula!

In fact, the answer to the “Completed Square” equation is the same as the Quadratic Formula.

But you are required to show the steps to completing the square.

Let’s look at an example below.

Example (Solve Quadratic Equations by Completing the Square)

2x2 – 3x – 9 = 0

Step 1: Change coefficient of x2 equal to 1

x2 – 3x/2 – 9/2 = 0

Step 2: Leave x2 and x terms on the Left Hand Side (LHS)

x2 – 3x/2 =  9/2

Step 3: Coefficient of x ÷2, square it, add to both sides

x2 – 3x/2 + (– 3/4)2 =  9/2 + (– 3/4)2

Step 4: Factorise the LHS of the equation

(x – 3/4)2 = 9/2 + 9/16

= 81/16

Step 5: Simplify to solve for x

(x – 3/4)2 = ± 9/4

x = 3/4 + 9/4  or x = 3/4 – 9/4

   = 3                      = –3/2 (ans)

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Secondary Math Revision Notes

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Solving Quadratic Equations by Completing the Square, math notes

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