PERFECT SQUARE TRINOMIALS

Example 1 :

Factor the following perfect square trinomial.

x² + 12x + 36

Solution :

= x² + 12x + 36

= x² + 2(x)(6) + 6²

The trinomial 'x² + 2(x)(6) + 6²' is in the form of

a² + 2ab + b²

Since a² + 2ab + b² = (a + b)²,

x² + 2(x)(6) + 6² = (x + 6)²

Therefore, the factors of 'x² + 12x + 36' are

(x + 6)(x + 6) or (x + 6)²

Example 2 :

Find the factors of the following perfect square trinomial.

a² + 4a + 4

Solution :

= a² + 4a + 4

= a² + 2(a)(2) + 2²

The trinomial 'a² + 2(a)(2) + 2²' is in the form of

a² + 2ab + b²

Since a² + 2ab + b² = (a + b)²,

a² + 2(a)(2) + 2² = (a + 2)²

Therefore, the factors of 'a² + 4a + 4' are

(a + 2)(a + 2) or (a + 2)²

Example 3 :

Check whether the following trinomial is a perfect square trinomial.

x² - 13x + 36

Solution :

The given trinomial is in the form of ax² + bx + c.

Comparing ax² + bx + c and x² - 13x + 36,

a = 1, b = -13 and c = 36

b² = (-13)² = 169

4ac = 4(1)(36) = 144

Since b² ≠ 4ac, the given trinomial is not a perfect square trinomial.

Example 4 :

Check whether the following trinomial is a perfect square trinomial. 

 x² - 10x + 100

Solution :

The given trinomial is in the form of ax² + bx + c.

Comparing ax² + bx + c and x² - 13x + 36,

a = 1, b = -10 and c = 100

b² = (-10)² = 100

4ac = 4(1)(100) = 400

Since b² ≠ 4ac, the given trinomial is not a perfect square trinomial.

Example 5 :

Check whether the following trinomial is a perfect square trinomial.

2x² + 17x + 21

Solution :

The given trinomial is in the form of ax² + bx + c.

Comparing ax² + bx + c and 2x² + 17x + 21,

a = 2, b = 17 and c = 21

b² = 17² = 289

4ac = 4(2)(21) = 168

Since b² ≠ 4ac, the given trinomial is not a perfect square trinomial.

Example 6 :

Check whether the following trinomial is a perfect square trinomial.

4x² + 8x + 4

Solution :

The given trinomial is in the form of ax² + bx + c.

Comparing ax² + bx + c and 4x² + 8x + 4,

a = 4, b = 8 and c = 4

b² = 8² = 64

4ac = 4(4)(4) = 64

Since b² = 4ac, the given trinomial is a perfect square trinomial.

Example 7 :

What value should be added to the following expression to make it as a perfect square?

x² + 6x

Solution :

In x² + 6x, write 6x as a multiple of 2. That is, in the form of 2ab.

x² + 6x = x² + 2(x)(3)

x² + 2(x)(3) is in the form of a² + 2ab.

We know that a² + 2ab + b² is a perfect square trinomial.

Comparing

x² + 2(x)(3)

and 

a² + 2ab + b²,

instead b², you must have +3² in  x² + 2(x)(3).

So, you have to add 3² or 9 to x² + 6x to make it as a perfect square.

Example 8 :

Solve :

x² - 8x + 16 = 0

Solution :

x² - 8x + 16 = 0

x² - 2(x)(4) + 16 = 0

x² - 2(x)(4) + 4² = 0

(x - 4)² = 0

Take square root on both sides.

x - 4 = 0

Add 4 to both sides.

x = 4

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