Factor. 7q^2 + 10q + 3 ____

Identify a, b, c: Identify a, b, and c in the quadratic expression 7q^2 + 10q + 3. Compare 7q^2 + 10q + 3 with ax^2 + bx + c. a = 7 b = 10 c = 3Find numbers multiply add: Find two numbers that multiply to a*c (7*3 = 21) and add up to b (10). We need to find two numbers that multiply to 21 and add up to 10. The numbers 7 and 3 satisfy these conditions because 7*3 = 21 and 7+3 = 10.Rewrite middle term: Rewrite the middle term 10q using the two numbers found in Step 2. We can express 10q as 7q + 3q. So, 7q^2 + 10q + 3 becomes 7q^2 + 7q + 3q + 3.Factor by grouping: Factor by grouping. Group the terms into two pairs: (7q^2 + 7q) and (3q + 3). Factor out the greatest common factor from each pair. From the first pair, we can factor out 7q: 7q(q + 1). From the second pair, we can factor out 3: 3(q + 1).Write factored form: Write the factored form of the expression. Since both groups contain the factor (q + 1), we can factor this out. The factored form is (7q + 3)(q + 1).

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Math Problems

Algebra 1

Factor quadratics with other leading coefficients

Full solution

Q. Factor.

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7q^2 + 10q + 3

Identify $a$ , $b$ , $c$ : Identify $a$ , $b$ , and $c$ in the quadratic expression $7q^2 + 10q + 3$ . Compare $7q^2 + 10q + 3$ with $ax^2 + bx + c$ . $a = 7$ $b$ $0$ $b$ $1$
Find numbers multiply add: Find two numbers that multiply to $a*c$ ( $7*3 = 21$ ) and add up to $b$ ( $10$ ). $\newline$ We need to find two numbers that multiply to $21$ and add up to $10$ . $\newline$ The numbers $7$ and $3$ satisfy these conditions because $7*3 = 21$ and $7+3 = 10$ .
Rewrite middle term: Rewrite the middle term $10q$ using the two numbers found in Step $2$ . $\newline$ We can express $10q$ as $7q + 3q$ . $\newline$ So, $7q^2 + 10q + 3$ becomes $7q^2 + 7q + 3q + 3$ .
Factor by grouping: Factor by grouping. $\newline$ Group the terms into two pairs: $7q^2 + 7q$ and $3q + 3$ . $\newline$ Factor out the greatest common factor from each pair. $\newline$ From the first pair, we can factor out $7q$ : $7q(q + 1)$ . $\newline$ From the second pair, we can factor out $3$ : $3(q + 1)$ .
Write factored form: Write the factored form of the expression. $\newline$ Since both groups contain the factor $(q + 1)$ , we can factor this out. $\newline$ The factored form is $(7q + 3)(q + 1)$ .