Perfect Square Trinomial
Perfect square trinomials are algebraic expressions with three terms that are obtained by multiplying a binomial with the same binomial. A perfect square is a number that is obtained by multiplying a number by itself. Binomials are algebraic expressions consisting of just two terms that are either separated by a positive (+) or a negative (-) sign. Similarly, trinomials are algebraic expressions consisting of iii terms. When a binomial consisting of a variable and a abiding is multiplied past itself, it results in a perfect square trinomial having three terms. The terms of a perfect square trinomial are separated by either a positive or a negative sign.
|1.||Perfect Square Trinomial Definition|
|2.||Perfect Square Trinomial Design|
|iii.||How to Factor a Perfect Square Trinomial?|
|four.||Perfect Square Trinomial Formula|
|v.||Perfect Square Trinomial FAQs|
Perfect Square Trinomial Definition
A perfect square trinomial is defined as an algebraic expression that is obtained past squaring a binomial expression. It is of the form ax2
+ bx + c. Here a, b, and c are real numbers and a ≠ 0. For example, let u.s.a. accept a binomial (x+iv) and multiply it to (x+4). The effect obtained is x2
+ 8x + xvi. A perfect square trinomial can be decomposed into ii binomials and the binomials when multiplied with each other gives the perfect square trinomial.
Perfect Square Trinomial Blueprint
Perfect square trinomial generally has the blueprint of aii
+ 2ab + b2or a2
– 2ab + b2. Given a binomial, to find the perfect square trinomial, we follow the steps given below. They are,
Notice the square the first term of the binomial.
Multiply the start and the 2nd term of the binomial with 2.
Discover the foursquare of the 2nd term of the binomial.
Sum up all the three terms obtained in steps one, 2, and three.
The first term of the perfect square trinomial is the square of the start term of the binomial. and the 2nd term is twice the product of the 2 terms of the binomial and the third term is the square of the 2nd term of the binomial. Accept a look at the figure shown below to sympathize the perfect square trinomial design. If the binomial being squared has a positive sign, and so all the terms in the perfect square trinomial are positive, whereas, if the binomial has a negative sign attached with its 2d term, then the second term of the trinomial (which is twice the product of the two variables) will exist negative.
How to Factor Perfect Square Trinomial?
Perfect square trinomials are either separated by a positive or a negative symbol between the terms. 2 important algebraic identities with regards to perfect foursquare trinomial are equally follows.
- (a + b)2
+ 2ab + bii
- (a – b)2
– 2ab + bii
The steps to be followed to factor a perfect square polynomial are equally follows.
- Write the given perfect square trinomial of the grade a2
+ 2ab + b2
– 2ab + b2, such that the first and the third terms are perfect squares, one being a variable and another being a constant.
- Check if the middle term is twice the product of the first and the 3rd term. Too, check the sign of the heart term.
- If the middle term is positive, then compare the perfect square trinomial with atwo
+ 2ab + b2
and if the eye term is negative, then compare the perfect square trinomial with a2
– 2ab + btwo.
- If the center 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 Foursquare Trinomial Formula
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 ax2+bx+c and also satisfies the condition of b2
= 4ac. At that place are two forms of a perfect square trinomial. They are,
- (ax)two+ 2abx + b2= (ax + b)two—– (1)
- (ax)2−2abx + b2
= (ax−b)two—– (two)
For case, allow us take a perfect square polynomial, ten2
+ 6x + 9. Comparing this with the grade axii+bx+c, we become a = one, b = 6 and c = ix. Let us cheque if this trinomial satisfies the condition btwo
= 36 and 4 × a × c = 4 × 1 × 9, which is equal to 36.
Therefore, the trinomial satisfies the condition b2
= 4ac. And so, we can telephone call information technology a perfect square trinomial.
Topics Related to Perfect Foursquare Trinomial
Cheque out some important topics related to perfect square trinomial.
- Factorization of Algebraic Expressions
- Squaring a Trinomial
- Perfect Square Trinomial Formula
Perfect Square Trinomial Examples
Gene the perfect square trinomial x2
+ 10x + 25.
We tin can see that the given expression tentwo
+ 10x + 25 is of the grade aii
+ 2ab + btwo. Factors of (atwo
+ 2ab + btwo) are (a+b) (a+b).
Hither, a = x and b = five and 2ab = 10x.
Therefore, the factors are (x + v) (x + 5) or (x + five)2.
Notice the factors of the perfect square trinomial 9xtwo
– 6x + 1.
Nosotros can see that the given equaion is of the form aii
– 2ab + bii. Factors of (a2
– 2ab + b2) are (a-b) (a-b).
Here, a = 3x, 2ab = 6x and b = one
Therefore, the factors of the perfect square trinomial are (3x -ane) (3x -1) or (3x -1)2.
Find if 4x2
+ 8x + 4 is a perfect foursquare trinomial or not.
To find if a given expression is a perfect square trinomial or non, we check if information technology satisfies the status btwo
Given equation = 4x2
+ 8x + 4.
Comparison this equation to ax2
+ bx + c
, we go a = 4, b = viii and c = 4.
= 8 × 8 (or) 64 —–(1)
4×a×c = 4 × iv × iv (or) 64 ——(2)
From (1) and (2), we find that b2
+ 8x + 4 is a perfect square trinomial.
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Practice Questions on Perfect Square Trinomial
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Perfect Foursquare Trinomial FAQs
What is a Perfect Square Trinomial?
A perfect square trinomial is an algebraic expression that is of the class ax2
+ bx + c, which has 3 terms. It is obtained by the multiplication of a binomial with itself. For example, ten2
+ 6x + 9 is a perfect square polynomial obtained past multiplying the binomial (x + three) by itself. In other words, (x +iii) (10 + three) = xii
+ 6x + nine.
How is Perfect Square Trinomial Formed?
A perfect square trinomial is formed by multiplying 2 binomials, which are ane and the same. A binomial is an algebraic expression with 2 terms and a trinomial is an algebraic expression with three terms. For example, if (a + ii) is a binomial, then the perfect square trinomial is obtained by multiplying (a+two) and (a+2), which gives a2
+ 4a + 4.
Should All the Three Terms in an Algebraic Expression Be Squares to Call it a Perfect Square Trinomial?
No, only the start and the third terms should be perfect squares for an algebraic expression to exist a perfect square trinomial, For example, in the expression xtwo
+ 2x + one, the first term is the square of ‘ten’ and the third term is the square of ‘ane’.
What is the Design of a Perfect Square Trinomial?
The perfect square trinomial is either of the pattern a2
+ 2ab + b2
– 2ab + btwo
. These expressions are obtained by squaring the binomials (a+b) and (a-b) respectively.
What are the Important Algebraic Expressions to be Known to Factor a Perfect Square Trinomial?
The important algebraic expressions to be known for factoring a perfect square trinomial are,
= a2+ 2ab + b2
= atwo−2ab + btwo
What is the Formula for Perfect Square Trinomial?
A perfect square trinomial is obtained by multiplying two aforementioned binomials. It takes the class of the following two expressions. They are,
- (ax)2+ 2abx + b2= (ax + b)two
- (ax)ii−2abx + b2
Also, an expression is said to be a perfect square trinomial if it is of the course axtwo+bx+c and b2
= 4ac is satisfied. For example, comparing the expression x2
+ 2x + one, to the forms mentioned to a higher place, we observe that a = ane, b = 2 and c = 1, therefore, b2
= four and 4 × a × c = 4 × one × ane, which is 4. Therefore, bii
= 4ac is satisfied and nosotros conclude that x2
+ 2x + i is a perfect square trinomial.
How do you Gene a Perfect Foursquare Trinomial?
A perfect square trinomial has three terms which may be in the form of (ax)2+ 2abx + b2= (ax + b)2
(or) (ax)2−2abx + bii
= (ax−b)2.The steps to exist followed to factor a perfect square polynomial are as follows.
- Verify whether the given perfect square trinomial is of the grade aii+ 2ab + b2
(or) a2−2ab + b2.
- Check if the middle term is twice the product of the starting time and the tertiary term. Likewise, check the sign of the heart term.
- If the center term is positive, then compare the perfect square trinomial with atwo
+ 2ab + b2
and if the middle term is negative, so compare the perfect square trinomial with a2
– 2ab + b2.
- If the eye term is positive, and then the factors are (a+b) (a+b) and if the heart term is negative, then the factors are (a-b) (a-b).
Tin can Perfect Square Trinomials be chosen Quadratic Equations?
Yes, perfect square polynomials can be of the class of quadratic equations. A quadratic equation consists of i squared term and it should have the caste of 2. Since perfect square trinomials accept their degree every bit two, we can call them quadratic equations. Note that all quadratic equations cannot be considered as perfect square trinomials every bit all quadratic equations do non satisfy the conditions required for a perfect square trinomial.
Can All Algebraic Expressions that take the Kickoff and the Last Terms as Perfect Squares exist chosen Perfect Square Trinomials?
No, not all the algebraic expressions that take the showtime and the terminal term as perfect squares be called perfect square trinomials. Let us sympathise this with an example, x2
+ 18x + 36 is an algebraic expression with three terms and the outset and third terms are perfect squares. In that case, comparing this expression to a2
+ 2ab + bii
we become a = 1 and b = half-dozen, then 2ab should be 12x for this expression to be a perfect foursquare trinomial. But hither the value of 2ab = 18x, therefore, we cannot say that all algebraic expressions with first and third terms as squares are perfect square trinomials.
Which Will Result in a Perfect Square Trinomial