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Solve by completing the square


Solve by completing the square could take a little bit more time to do than solving by factoring. However, the steps are straightforward.

Before showing examples, you need to understand what a perfect square trinomial is

Binomial × same Binomial = Perfect square trinomial

(x + 4) × (x + 4) = x2 + 4x + 4x + 16 = x2 + 8x + 16

(x + a) × (x + a) = x2 + ax + ax + a2 = x2 + 2ax + a2

Important observation:

What is the relationship between the coefficient of the second term and the last term?

For x2 + 8x + 16, coefficient of the second term is 8 and the last term is 16

(8/2)2 = 42 = 16

For, x2 + 2ax + a2, the coefficient of the second term is 2a and the last term is a2

(2a/2)2 = a2

So, what is the relationship?

The last term is obtained by dividing the coefficient of the second term by 2 and squaring the result

Now, let's say you have x2 + 20x and you want to find the last term to make the whole thing a perfect square trinomial

Just do (20/2)2 = 102. The perfect square trinomial is x2 + 20x + 102= (x + 10) × (x + 10)

Example #1:

Solve by completing the square x2 + 6x + 8 = 0

x2 + 6x + 8 = 0

Subtract 8 from both sides of the equation.

x2 + 6x + 8 - 8 = 0 - 8

x2 + 6x = - 8

To complete the square, always do the following 24 hours a day 365 days a year. It is never going to change when you solve by completing the square!

You are basically looking for a term to add to x2 + 6x that will make it a perfect square trinomial.

To this end, get the coefficient of the second term, divide it by 2 and raise it to the second power.

The second term is 6x and the coefficient is 6.

6/2 = 3 and after squaring 3, we get 32

x2 + 6x = - 8

Add 32 to both sides of the equation above

x2 + 6x + 32 = - 8 + 32

x2 + 6x + 32 = - 8 + 32

(x + 3)2 = -8 + 9

(x + 3)2 = 1

Take the square root of both sides

√((x + 3)2) = √(1)

x + 3 = ±1

When x + 3 = 1, x = -2

When x + 3 = -1, x = -4

Example #2:

Solve by completing the square x2 + -6x + 8 = 0 instead of x2 + 6x + 8 = 0

The second term this time is -6x and the coefficient is -6.

-6/2 = -3 and after squaring -3, we get (-3)2 = 9

x2 + -6x = - 8

Add (-3)2 to both sides of the equation above

x2 + -6x + (-3)2 = - 8 + (-3)2

(x + -3)2 = -8 + 9

(x + -3)2 = 1

Take the square root of both sides

√((x + -3)2) = √(1)

x + -3 = ±1

When x + -3 = 1, x = 4

When x + -3 = -1, x = 2

Example #3:

Solve by completing the square 3x2 + 8x + -3 = 0

3x2 + 8x + -3 = 0

Divide everything by 3. Always do that when the coefficient of the first term is not 1

(3/3)x2+ (8/3)x + -3/3 = 0/3

x2+ (8/3)x + -1 = 0

Add 1 to both sides of the equation.

x2 + (8/3)x + -1 + 1 = 0 + 1

x2 + (8/3)x = 1

The second term is (8/3)x and the coefficient is 8/3.

8/3 ÷ 2 = 8/3 × 1/2 = 8/6 and after squaring 8/6, we get (8/6)2

x2 + (8/3)x = 1

Add (8/6)2 to both sides of the equation above

x2 + (8/3)x + (8/6)2 = 1 + (8/6)2

(x + 8/6)2 = 1 + 64/36

(x + 8/6)2 = 36/36 + 64/36 = (36 + 64)/36 = 100/36

Take the square root of both sides

√((x + 8/6)2) = √(100/36)

x + 8/6 = ±10/6

x + 8/6 = 10/6

x = 10/6 - 8/6 = 2/6 = 1/3

x + 8/6 = - 10/6

x = -10/6 - 8/6 = -18/6 = -3



To solve by completing the square can become quickly hard as shwon in example #3







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