Factor completely. 7+14 x+7x^(2)=

Recognize the structure: Recognize the structure of the expression. The given expression is a quadratic in the form of ax^2 + bx + c. We need to factor it completely.Look for common factors: Look for common factors in all terms. All terms in the expression 7 + 14x + 7x^2 have a common factor of 7. Factor out the common factor of 7. 7(1 + 2x + x^2)Factor out the common factor: Recognize the structure of the expression inside the parentheses. The expression inside the parentheses is a perfect square trinomial because it can be written in the form (a + b)^2 where a^2 = x^2, 2ab = 2x, and b^2 = 1.Recognize the structure inside: Factor the perfect square trinomial. The expression 1 + 2x + x^2 can be factored as (x + 1)^2. So, the factored form of the expression inside the parentheses is (x + 1)(x + 1).Factor the perfect square trinomial: Write the final factored form of the original expression. The final factored form of the expression 7 + 14x + 7x^2 is 7(x + 1)^2.

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Math Problems

Algebra 1

Factor quadratics: special cases

Full solution

Q. Factor completely.

\newline

7+14x+7x^{2}=

Recognize the structure: Recognize the structure of the expression. $\newline$ The given expression is a quadratic in the form of $ax^2 + bx + c$ . We need to factor it completely.
Look for common factors: Look for common factors in all terms. $\newline$ All terms in the expression $7 + 14x + 7x^2$ have a common factor of $7$ . $\newline$ Factor out the common factor of $7$ . $\newline$ $7(1 + 2x + x^2)$
Factor out the common factor: Recognize the structure of the expression inside the parentheses. $\newline$ The expression inside the parentheses is a perfect square trinomial because it can be written in the form $(a + b)^2$ where $a^2 = x^2$ , $2ab = 2x$ , and $b^2 = 1$ .
Recognize the structure inside: Factor the perfect square trinomial. $\newline$ The expression $1 + 2x + x^2$ can be factored as $(x + 1)^2$ . $\newline$ So, the factored form of the expression inside the parentheses is $(x + 1)(x + 1)$ .
Factor the perfect square trinomial: Write the final factored form of the original expression. $\newline$ The final factored form of the expression $7 + 14x + 7x^2$ is $7(x + 1)^2$ .