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# Perfect Square Trinomial – Explanation & Examples

A quadratic equation is a polynomial of second degree usually in the form of f(x) = ax^{2} + bx + c where a, b, c, ∈ R and a ≠ 0. The term ‘a’ is referred to as the leading coefficient, while ‘c’ is referred to as the absolute term of f (x).

Every quadratic equation has two values of the unknown variable usually known as the roots of the equation (α, β). The roots of a quadratic equation can be obtained by factoring the equation.

## What is a Perfect Square Trinomial?

The ability to **recognize special cases of polynomials** that can be easily factored is a fundamental skill for solving any algebraic expressions that involve polynomials.

One of these “**easy to factor**” polynomials is the perfect square trinomial. We can recall that a trinomial is an algebraic expression composed of three terms that are connected by addition or subtraction.

Similarly, a binomial is an expression **composed of two terms**. Therefore, a perfect square trinomial can be defined as an expression that is obtained by squaring a binomial

Learning** how to recognize a perfect square trinomial** is the first step to factoring it.

*The following are the tips on how to recognize a perfect square trinomial:*

- Check whether the first and last terms of the trinomial are perfect squares
- Multiply the roots of the first and third terms together.
- Compare to the middle terms with the result in step two
- If the first and last terms are perfect squares, and the coefficient of the middle term is twice the product of the square roots of the first and last terms, then, the expression is a prefect square trinomial.

## How to Factor a Perfect Square Trinomial?

Once you have identified a perfect square trinomial, factoring it is quite a straightforward process.

*Let’s take a look at the steps for factoring a perfect square trinomial.*

- Identify the squared numbers in the first and third terms of the trinomial
- Examine the middle term if it has either positive or negative. If the middle term of the trinomial is positive or negative, then the factors will have a plus and minus sign respectively.
- Write out your terms by applying the following identities:

(i) a^{2} + 2ab + b^{2} = (a + b)^{2} = (a + b) (a + b)

(ii) a^{2} – 2ab + b^{2} = (a – b)^{2} = (a – b) (a – b)

### Perfect Square Trinomial Formula

An expression obtained from the square of binomial equation is a perfect square trinomial. An expression is said to a perfect square trinomial if it takes the form ax^{2} + bx + c and satisfies the condition b^{2} = 4ac.

*The perfect square formula takes the following forms:*

- (ax)
^{2 }+ 2abx + b^{2 }= (ax + b)^{2} - (ax)
^{2 }−2abx + b^{2}= (ax−b)^{2}

*Example 1*

Factor x^{2}+ 6x + 9

__Solution__

We can rewrite the expression x^{2} + 6x + 9 in the form a^{2} + 2ab + b^{2 }as;

x^{2}+ 6x + 9 ⟹ (x)^{2} + 2 (x) (3) + (3)^{2}

Applying the formula of a^{2} + 2ab + b^{2} = (a + b)^{2} to the expression gives;

= (x + 3)^{2}

= (x + 3) (x + 3)

*Example 2*

Factor x^{2} + 8x + 16

__Solution__

Write the expression x^{2} + 8x + 16 as a^{2} + 2ab + b^{2}

x^{2} + 8x + 16 ⟹ (x)^{2} + 2 (x) (4) + (4)^{2}

Now we will apply the perfect square trinomial formula;

= (x + 4)^{2}

= (x + 4) (x + 4)

*Example 3*

Factor 4a^{2} – 4ab + b^{2}

__Solution__

4a^{2} – 4ab + b^{2} ⟹ (2a)^{2} – (2)(2) ab + b^{2}

= (2a – b)^{2}

= (2a – b) (2a – b)

*Example 4*

Factor 1- 2xy- (x^{2} + y^{2})

__Solution__

1- 2xy- (x^{2} + y^{2})

= 1 – 2xy – x^{2} – y^{2}

= 1 – (x^{2} + 2xy + y^{2})

= 1 – (x + y )^{2}

= (1)^{2} – (x + y)^{2}

= [1 + (x + y)] [1 – (x + y)]

= [1 + x + y] [1 – x – y]

*Example 5*

Factor 25y^{2} – 10y + 1

__Solution__

25y^{2} – 10y + 1⟹ (5y)^{2} – (2)(5)(y)(1) + 1^{2}

= (5y – 1)^{2}

= (5y– 1) (5y – 1)

*Example 6*

Factor 25t^{2} + 5t/2 + 1/16.

__Solution__

25t^{2} + 5t/2 + 1/16 ⟹ (5t)^{2 }+ (2)(5)(t) (1/4) + (1/4)^{2}

= (5t + 1/4)^{2}

= (5t + 1/4) (5t + 1/4)

*Example 7*

Factor x^{4} – 10x^{2}y^{2} + 25y^{4}

__Solution__

x^{4} – 10x^{2}y^{2} + 25y^{4} ⟹ (x^{2})^{2} – 2 (x^{2}) (5y^{2}) + (5y^{2})^{2}

Apply the formula a^{2} + 2ab + b^{2} = (a + b)^{2} to get,

= (x^{2} – 5y^{2})^{2}

= (x^{2} – 5y^{2}) (x^{2} – 5y^{2})

*Practice Questions*

Factorize the following perfect square trinomials:

- x
^{2 }+ 12x + 36 - 9a
^{2 }– 6a + 1 - (m + n)
^{2 }+ 12(m + n) + 36 - x
^{2 }+ 4x + 4 - x
^{2}+ 2x + 1 - x
^{2}+ 10x + 25 - 16x
^{2}– 48x + 36 - x
^{2}+ x + ¼ - Z
^{2}+ 1/z^{2}– 2. - 4x
^{2}– 20x + 25

__Answers__

- (x + 6) (x + 6)
- (3a – 1) (3a – 1)
- (m + n + 6) (m + n + 6)
- (x + 2) (x + 2)
- (x + 1) (x + 1)
- (x + 5) (x + 5)
- (4x– 6) (4x – 6)
- (x + 1/2) (x + 1/2)
- (z – 1/z
^{2}) (z – 1/z^{2}) - (2x – 5) (2x – 5)

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