a – b)2 = (a – b)(a – b) = a2 – 2ab + b2 Find the product of (k – 9)2 using the perfect square trinomial rule shown on the lef
The Correct Answer and Explanation is:
To find the product of (k−9)2(k – 9)^2(k−9)2, we apply the perfect square trinomial rule:(a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2(a−b)2=a2−2ab+b2
In this case:
- a=ka = ka=k
- b=9b = 9b=9
Substitute these values into the rule:(k−9)2=k2−2(k)(9)+92(k – 9)^2 = k^2 – 2(k)(9) + 9^2(k−9)2=k2−2(k)(9)+92
Now simplify each term:
- k2k^2k2 remains as is
- 2(k)(9)=18k2(k)(9) = 18k2(k)(9)=18k
- 92=819^2 = 8192=81
So:(k−9)2=k2−18k+81(k – 9)^2 = k^2 – 18k + 81(k−9)2=k2−18k+81
Explanation
The expression (k−9)2(k – 9)^2(k−9)2 is an example of a binomial squared. This means a two-term expression is multiplied by itself. When you square a binomial, you can expand it either by multiplying it manually or by using the perfect square trinomial rule. This rule simplifies the process and helps avoid mistakes during expansion.
The perfect square trinomial rule is based on a pattern observed when squaring binomials. Specifically, for any real numbers aaa and bbb, the square of the expression (a−b)(a – b)(a−b) follows the pattern:(a−b)2=a2−2ab+b2(a – b)^2 = a^2 – 2ab + b^2(a−b)2=a2−2ab+b2
This pattern includes three parts: the square of the first term, minus two times the product of the first and second terms, plus the square of the second term. Recognizing and using this pattern allows for quick and accurate expansion.
In our case, kkk is the first term and 999 is the second. Squaring kkk gives k2k^2k2, multiplying 2×k×92 \times k \times 92×k×9 gives 18k18k18k, and squaring 999 gives 818181. Placing these in the pattern gives the trinomial:k2−18k+81k^2 – 18k + 81k2−18k+81
This trinomial is the product of the binomial squared. Understanding and applying this rule helps in algebraic simplification, solving equations, and factoring. It is one of the foundational identities used throughout algebra.
