# 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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To solve the quadratic equation ax^{2} + bx + c = 0 by completing the square, you can follow the steps below:

**Step 1: Change coefficient of x ^{2} equal to 1**

a( *x*^{2} + b*x*/a + c/a) = 0

**Step 2: Leave x ^{2} and x terms on the Left Hand Side (LHS)**

*x*^{2} + b*x*/a = – c/a

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

*x*^{2} + b*x*/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/4a^{2} + b^{2}/4a^{2}

= (b^{2} – 4ac)/4a^{2}

x + b/2a = ±√( (b^{2} – 4ac)/4a^{2 })

x = – b/2a ±√( (b^{2} – 4ac)/4a^{2 })

x =( – b ±√( (b^{2} – 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)**

2*x*^{2} – 3*x* – 9 = 0

**Step 1: Change coefficient of x ^{2} equal to 1**

*x*^{2} – 3*x/2* – 9/2 = 0

**Step 2: Leave x ^{2} and x terms on the Left Hand Side (LHS)**

*x*^{2} – 3*x*/2 = 9/2

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

*x*^{2} – 3*x*/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)*

## Last Minute Revision for O Level Math?

## Check out our exam guide on other topics here!

**Before you go, 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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