How to Solve Perfect Square Trinomial?

A perfect square trinomial is formed by multiplying two binomials, which are the same. In this step-by-step guide, you will become more familiar with solving perfect square trinomial.

How to Solve Perfect Square Trinomial?

Perfect square trinomials are algebraic expressions with three terms that are obtained by multiplying a binomial with the same binomial.

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Step by Step guide to perfect square trinomial

A perfect square trinomial is defined as an algebraic expression obtained by squaring a binomial expression.  It is of the form \(ax^2+ bx+ c\). Here \(a, b,\) and \(c\) are real numbers and \(a≠0\).

A perfect square trinomial can be decomposed into two binomials and the binomials give the perfect square trinomial when multiplied by each other.

Perfect square trinomial pattern

Perfect square trinomial generally has the pattern of \(a^2+ 2ab + b^2\) or \(a^2 – 2ab + b^2\). According to a binomial, to find the perfect square trinomial, follow the steps below:

  • Step 1: Find the square of the first term of the binomial.
  • Step 2: Multiply the first and the second term of the binomial with \(2\).
  • Step 3: Find the square of the second term of the binomial.
  • Step 4: Sum up all the three terms obtained in steps \(1, 2,\) and \(3\).

The first term of the perfect square trinomial is the square of the first term of the binomial. And the second term is twice the product of the two terms of the binomial and the third term is the square of the second term of the binomial.

To understand the perfect square trinomial pattern, look at the figure below. If the binomial being squared has a positive sign, then all the terms in the perfect square trinomial are positive, while, if the binomial has a negative sign attached with its second term, then the second term of the trinomial (which is twice the product of the two variables) will be negative.

How to factor perfect square trinomial?

Perfect square trinomials are either separated by a positive or a negative sign between the terms. Two important algebraic identities with regards to perfect square trinomial are as follows:

  • \(\color{blue}{(a + b)^2= a^2+ 2ab + b^2}\)
  • \(\color{blue}{(a-b)^2 = a^2 – 2ab + b^2}\)

The steps to follow for factoring a perfect square polynomial are as follows:

  • Step 1: Write the given perfect square trinomial of the form \(a^2+ 2ab + b^2\) or \(a^2 – 2ab + b^2\), so that the first and the third terms are perfect squares, one being a variable and another being a constant.
  • Step 2: Check whether the middle term is twice the product of the first and the third term. Also, check the sign of the middle term.
  • Step 3: If the middle term is positive, then compare the perfect square trinomial with \(a^2+ 2ab + b^2\) and if the middle term is negative, then compare the perfect square trinomial with \(a^2 – 2ab + b^2\).
  • Step 4: If the middle term is positive, then the factors are \((a+b) (a+b)\) and if the middle term is negative, then the factors are \((a-b) (a-b)\).

Perfect square trinomial formula

A perfect square trinomial is obtained by multiplying the same binomial expression with each other.  A trinomial is said to be a perfect square if it is of the form \(ax^2+bx+c\) and also satisfies the condition of \(b^2= 4ac\). There are two forms of a perfect square trinomial:

  • \(\color{blue}{(ax)^2+ 2abx + b^2= (ax + b)^2}\)
  • \(\color{blue}{(ax)^2−2abx + b^2= (ax−b)^2}\)

For example, let us take a perfect square polynomial, \(x^2+ 10x + 25\). Comparing this with the form \(ax^2+bx+c\), we get \(a= 1, b= 10\) and \(c = 25\). Let us check if this trinomial satisfies the condition \(b^2= 4ac\).
\(b^2= 100\) and \(4× a × c = 4 × 1 × 25\), which is equal to \(100\).

Therefore, the trinomial satisfies the condition \(b^2 = 4ac\). So, we can call it a perfect square trinomial.

Perfect Square Trinomial – Example 1:

Find the factors of the perfect square trinomial \(9x^2- 6x + 1\).

Solution:

The given equaion is of the form \(a^2- 2ab + b^2\) and factors of \((a^2 – 2ab + b^2)\) are \((a-b) (a-b)\).

Here, \(a = 3x\), \(2ab = 6x\) and \(b = 1\). So, the factors of the perfect square trinomial are \((3x -1) (3x -1)\) or \((3x -1)^2\).

Exercises for Perfect Square Trinomial

Factor each Trinomial.

  1. \(\color{blue}{x^2+8x+16}\)
  2. \(\color{blue}{x^2+24x+144}\)
  3. \(\color{blue}{x^2-18x+81}\)
  4. \(\color{blue}{x^2+4x+4}\)
  5. \(\color{blue}{4x^2-28x+49}\)
This image has an empty alt attribute; its file name is answers.png
  1. \(\color{blue}{\left(x+4\right)^2}\)
  2. \(\color{blue}{\left(x+12\right)^2}\)
  3. \(\color{blue}{\left(x-9\right)^2}\)
  4. \(\color{blue}{\left(x+2\right)^2}\)
  5. \(\color{blue}{\left(2x-7\right)^2}\)

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