Perfect Square Trinomial Formula
Perfect Square Trinomial Formula
Perfect square trinomials are algebraic expressions containing three terms and obtained by multiplying two binomials by the same binomial. Numbers that are perfect squares are obtained by multiplying them by themselves. An algebraic expression containing only two terms is called a binomial. It consists of a positive (+) or a negative (-) sign between the terms. Similarly, trinomials are algebraic expressions with three terms. The perfect square trinomial has three terms when a binomial consists of a variable and a constant is multiplied by itself. There is either a positive or a negative sign separating the terms in a Perfect Square Trinomial Formula. Perfect Square Trinomial Formula can be understood to its complete depth with the help of study resources provided by Extramarks. Perfect Square Trinomial Formula and many other important concepts can be studied thoroughly with the help of the expert guidance of Extramarks’ educators.
Perfect Square Trinomial Definition
A Perfect Square Trinomial Formula are algebraic expressions obtained by squaring binomial expressions. It is of form ax2 + bx + c. The numbers a, b, and c are real numbers, while a is not equal to 0. Let’s take a binomial (x+4) and multiply it by (x+4). The result is x2 + 8x + 16. The Perfect Square Trinomial Formula can be decomposed into two binomials, which when multiplied together yield the perfect square trinomial.
Perfect Square Trinomial Pattern
Perfect square trinomial generally has a2 + 2ab + b2 or a2 – 2ab + b2. To find the Perfect Square Trinomial Formula for a binomial, follow the steps below. They are,
- The first step is to find the square, or the first term of the binomial.
- The second step is to multiply the first and second terms of the binomial with 2.
- The third step is to find the square of the binomial’s second term.
- The fourth step is to add up all three terms obtained in steps 1, 2, and 3.
The square of the first term of the binomial is the first term of the Perfect Square Trinomial Formula. In the binomial, the second term is twice the product of the two terms, and the third term is the square of the second term. It follows that all terms of the Perfect Square Trinomial Formula are positive if the binomial being squared has a positive sign. The second term of the trinomial will be negative if its second term is negative (which is twice the product of the two variables).
Perfect Square Trinomial Formula
There is either a positive or a negative symbol between the terms of perfect square trinomials. There are two important algebraic identities related to the Perfect Square Trinomial Formula.
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
To factor a perfect square polynomial, follow these steps.
- Write the Perfect Square Trinomial Formula of the form a2 + 2ab + b2 or a2 – 2ab + b2, such that the first and the third terms are perfect squares, one being a variable and another a constant.
- Make sure that the middle term is twice the product of the first and third terms. Be sure to check the sign of the middle term as well.
- If the middle term is positive, then the perfect square trinomial should be compared to a2 + 2ab + b2, and if the middle term is negative, then the Perfect Square Trinomial Formula should be compared to a 2 – 2ab + b2.
- In case of a positive middle term, the factors are (a+b) (a+b), and in case of a negative middle term, the factors are (a-b) (a-b).
Topics Related to Perfect Square Trinomial
Multiplying the same binomial expression with each other yields the Perfect Square Trinomial Formula. When a trinomial is of the form ax 2 + bx + c and also meets the condition b 2 = 4ac, it is said to be a perfect square. A Perfect Square Trinomial Formula can be expressed in two ways. They are,
- (ax)2+ 2abx + b2= (ax + b)2—– (1)
- (ax)2−2abx + b2 = (ax−b)2—– (2)
For example, take a perfect square polynomial, x2 + 6x + 9. Comparing this with ax2+bx+c, a = 1, b = 6 and c = 9. Check if trinomial satisfies the condition b2 = 4ac.
b2 = 36 and 4 × a × c = 4 × 1 × 9, which is equal to 36.
Due to this, the trinomial satisfies the condition b2 = 4ac. Therefore, it is called a Perfect Square Trinomial Formula. Take a look at some important topics related to perfect square trinomials.
- Algebra is the study of representations of problems and situations in the form of mathematical expressions. In order to form a meaningful mathematical expression, variables like x, y, and z must be used along with mathematical operations such as addition, subtraction, multiplication, and division.
- Arithmetic is used in all branches of mathematics, such as trigonometry, calculus, and coordinate geometry. The expression 2x + 4 = 8 is a simple example of an algebraic expression.
- It is possible to factorize algebraic expressions by finding the factors of the given expression, which means finding two or more expressions whose product is the given expression. It is the process of finding two or more expressions whose product is the given expression that is known as factorization of algebraic expressions. When a number divides a given number without leaving a remainder, it is called a factor. Multiplying two numbers means expressing a number as a result of their multiplication. It is common for algebraic expressions to be written as products of their factors. In algebraic expressions, numbers and variables are combined with arithmetic operations like addition and subtraction.
- The square of a trinomial has been discussed. As an example, if one is asked to find the square of 8, you would immediately say 64, since 8 × 8 = 64.
- A Perfect Square Trinomial Formula can be used to solve complex trinomial functions. Perfect Square Trinomial Formula functions are obtained by squaring binomial expressions. In order to be a perfect square, a trinomial must be of the form ax 2 + bx + c and satisfy the condition b 2 = 4ac. The Perfect Square Trinomial Formula can be understood by looking at solved examples.
Perfect Square Trinomial Examples
- Example 1: Factor the trinomial x2 + 10x + 25.
Solution:
The given expression x2 + 10x + 25 is of the form a2 + 2ab + b2. Factors of (a2 + 2ab + b2) are (a+b) (a+b).
Therefore, the factors are (x + 5) (x + 5) or (x + 5)2.
Practice Questions on Perfect Square Trinomial
Q1. Which of the following forms does a perfect square trinomial take?
Responses
- (ax)2 + 2abx + b2 = (ax + b)2
- (ax)2 −2abx + b2 = (ax−b)2
- Both (a) and (b)
- None of the above
FAQs (Frequently Asked Questions)
1. Perfect Square Trinomials: What are they?
There are three terms in the Perfect Square Trinomial Formula, which is of the form ax2 + bx + c. The formula is obtained by multiplying a binomial by itself. The Perfect Square Trinomial Formula x2 + 6x + 9 can be obtained by multiplying the binomial (x + 3) by itself. To put it another way, (x + 3) (x + 3) = x2 + 6x + 9.
2. What is the formula for the Perfect Square Trinomial?
By multiplying two binomials, which are one and the same, a perfect square trinomial is formed. An algebraic expression with two terms is a binomial, and an algebraic expression with three terms is a trinomial. The Perfect Square Trinomial Formula can be obtained by multiplying (a+2) and (a+2), which gives 2a2 + 4a + 4.
3. To call an algebraic expression a perfect square trinomial, must all three terms be squares?
For an algebraic expression to be a Perfect Square Trinomial Formula, only the first and third terms must be perfect squares. For example, in x 2 + 2x + 1, the first term is the square of ‘x’, and the third term is the square of ‘1’.
4. Perfect Square Trinomials: What is their pattern?
There are two possible patterns for the Perfect Square Trinomial Formula: a 2 + 2 ab + b 2 or a 2a2 – 2 ab + b 2. By squaring the binomials (a+b) and (a-b), one obtains these expressions.
5. How to Factor a Perfect Square Trinomial?
A Perfect Square Trinomial Formula has three terms which may be of the form (ax)2+ 2abx + b2= (ax + b)2 or (ax)2−2abx + b2 = (ax−b)2. Factoring a perfect square polynomial involves the following steps.
- One must verify that the given perfect square trinomial is of the form a 2 + 2ab + b 2.
- The middle term should be twice the product of the first and third terms. Be sure to check the middle term’s sign as well.
- The perfect square trinomial can be compared with a2 + 2ab + b 2 if the middle term is positive, and a2 – 2ab + b2 if the middle term is negative.
- In case of a positive middle term, the factors are (a+b) (a+b), and in case of a negative middle term, the factors are (a-b) (a-b).
6. Is it possible to call perfect square trinomials quadratic equations?
Quadratic equations can be formed from perfect square polynomials. Quadratic equations have one squared term and a degree of 2. Since perfect square trinomials have a degree of 2, we can call them quadratic equations. As all quadratic equations do not satisfy the conditions for a perfect square trinomial, they cannot be considered perfect square trinomials.
7. Would all algebraic expressions that have perfect squares as the first and last terms qualify as perfect square trinomials?
No, not all algebraic expressions with perfect squares in the first and last terms are perfect square trinomials. As an example, consider x2 + 18x + 36 which is an algebraic expression with three terms, and the first and third terms are perfect squares. To be a perfect square trinomial, we should have 2ab equal 12x when compared to a 2 + 2ab + b 2. Based on that, a = 1 and b = 6, so this expression is a perfect square trinomial. In this case, however, 2ab = 18x, so one cannot say that all algebraic expressions with first and third terms squared are perfect square trinomials.
