Question

question 20 of 25
what is the factorization of the expression below?
$x^2 - 14x + 49$

a. $(x + 7)(x - 7)$
b. $(x + 1)(x - 13)$
c. $(x - 1)(x - 13)$
d. $(x - 7)(x - 7)$

Explanation:

Step1: Recall perfect square formula

The perfect square trinomial formula is \(a^2 - 2ab + b^2=(a - b)^2=(a - b)(a - b)\).

Step2: Identify \(a\) and \(b\) in the given expression

For the expression \(x^2-14x + 49\), we have \(a=x\) (since \(a^2=x^2\)), and \(2ab = 14x\). Substituting \(a = x\) into \(2ab=14x\), we get \(2\times x\times b=14x\), so \(b = 7\) (because \(2xb=14x\Rightarrow b = 7\)). Also, \(b^2=49\) which matches the constant term.

Step3: Apply the formula

Using the perfect square trinomial formula \(a^2-2ab + b^2=(a - b)(a - b)\), with \(a=x\) and \(b = 7\), we get \(x^2-14x + 49=(x - 7)(x - 7)\). We can also check by expanding the options:

• Option A: \((x + 7)(x - 7)=x^2-49

eq x^2-14x + 49\)

• Option B: \((x + 1)(x - 13)=x^2-13x+x - 13=x^2-12x - 13

eq x^2-14x + 49\)

• Option C: \((x - 1)(x - 13)=x^2-13x-x + 13=x^2-14x + 13

eq x^2-14x + 49\)

• Option D: \((x - 7)(x - 7)=x^2-7x-7x + 49=x^2-14x + 49\)

Answer:

D. \((x - 7)(x - 7)\)