Factor. 3d^2 + 22d + 7 ____

Identify a, b, c: Identify a, b, and c in the quadratic expression 3d^2 + 22d + 7 by comparing it with the standard form ax^2 + bx + c. a = 3 b = 22 c = 7Find Multiplying Numbers: Find two numbers that multiply to a*c (3*7=21) and add up to b (22). We need to find two numbers that multiply to 21 and add up to 22. The numbers 1 and 21 satisfy these conditions because 1*21 = 21 and 1+21 = 22.Rewrite Middle Term: Rewrite the middle term, 22d, using the two numbers found in the previous step. 3d^2 + 22d + 7 can be rewritten as 3d^2 + 1d + 21d + 7 by splitting the middle term.Group Terms for Factoring: Group the terms to factor by grouping. (3d^2 + 1d) + (21d + 7)Factor Out Common Factors: Factor out the greatest common factor from each group. From the first group, we can factor out d: d(3d + 1) From the second group, we can factor out 7: 7(3d + 1)Final Factored Form: Since both groups contain the common factor (3d + 1), we can factor this out. The factored form is (d + 7)(3d + 1).

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Math Problems

Algebra 1

Factor quadratics with other leading coefficients

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Q. Factor.

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3d^2 + 22d + 7

Identify $a$ , $b$ , $c$ : Identify $a$ , $b$ , and $c$ in the quadratic expression $3d^2 + 22d + 7$ by comparing it with the standard form $ax^2 + bx + c$ .
$a = 3$
$b = 22$
$b$ $0$
Find Multiplying Numbers: Find two numbers that multiply to $a*c$ ( $3*7=21$ ) and add up to $b$ ( $22$ ). $\newline$ We need to find two numbers that multiply to $21$ and add up to $22$ . $\newline$ The numbers $1$ and $21$ satisfy these conditions because $1*21 = 21$ and $1+21 = 22$ .
Rewrite Middle Term: Rewrite the middle term, $22d$ , using the two numbers found in the previous step. $\newline$ $3d^2 + 22d + 7$ can be rewritten as $3d^2 + 1d + 21d + 7$ by splitting the middle term.
Group Terms for Factoring: Group the terms to factor by grouping. $\newline$ $(3d^2 + 1d) + (21d + 7)$
Factor Out Common Factors: Factor out the greatest common factor from each group. $\newline$ From the first group, we can factor out $d$ : $d(3d + 1)$ $\newline$ From the second group, we can factor out $7$ : $7(3d + 1)$
Final Factored Form: Since both groups contain the common factor $(3d + 1)$ , we can factor this out. $\newline$ The factored form is $(d + 7)(3d + 1)$ .