Examkin

Question

Difficulty: EasyFactoring Polynomials

Complete the factorization of the perfect square trinomial below by filling in the missing number.

Answer:x2+14x+49=(x+【7】)2x^2 + 14x + 49 = (x + 【7】)^2

Answer

7
A perfect square trinomial takes the general form x2+2bx+b2=(x+b)2x^2 + 2bx + b^2 = (x + b)^2. In the given expression x2+14x+49x^2 + 14x + 49, the constant term is 4949, which is 727^2. The middle term is 14x14x, which is equal to 2×7×x2 \times 7 \times x. Therefore, the expression factors directly as (x+7)2(x + 7)^2, meaning the missing number in the blank is 7.

Step-by-Step Solution

1
Recognize the polynomial form
The expression x2+14x+49x^2 + 14x + 49 matches the structure of a perfect square trinomial: a2+2ab+b2a^2 + 2ab + b^2.
The first term is a perfect square (x2x^2), and the last term is a perfect square (49=7249 = 7^2).
2
Verify the middle term
Checking the middle term: 2×x×7=14x2 \times x \times 7 = 14x. This matches the middle term of the given expression.
Confirming the middle term proves the expression can be factored exactly as (a+b)2(a + b)^2.
3
Determine the missing value
The expression factors as (x+7)2(x + 7)^2.
Using the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2 where a=xa = x and b=7b = 7.

Key Concept

Factoring perfect square trinomials
Rate this question

Topics

Factoring Polynomials

More questions from this topic