Factor. 2t^2 + 5t + 3 ____

Identify Coefficients: Identify the coefficients a, b, and c in the quadratic expression 2t^2 + 5t + 3 by comparing it to the standard form ax^2 + bx + c. a = 2, b = 5, c = 3Find Multiplying Numbers: Find two numbers that multiply to a*c (2*3 = 6) and add up to b (5). The numbers that satisfy these conditions are 2 and 3 because 2*3 = 6 and 2+3 = 5.Rewrite Middle Term: Rewrite the middle term (5t) using the two numbers found in the previous step. 2t^2 + 5t + 3 can be rewritten as 2t^2 + 2t + 3t + 3.Factor by Grouping: Factor by grouping. Group the first two terms and the last two terms. (2t^2 + 2t) + (3t + 3)Factor out Common Factor: Factor out the greatest common factor from each group. 2t(t + 1) + 3(t + 1)Check Factored Form: Since both groups contain the common factor (t + 1), factor it out. (2t + 3)(t + 1)Check the factored form by expanding it to ensure it equals the original expression. (2t + 3)(t + 1) = 2t^2 + 2t + 3t + 3 = 2t^2 + 5t + 3

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Math Problems

Algebra 1

Factor quadratics with other leading coefficients

Full solution

Q. Factor.

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2t^2 + 5t + 3

Identify Coefficients: Identify the coefficients $a$ , $b$ , and $c$ in the quadratic expression $2t^2 + 5t + 3$ by comparing it to the standard form $ax^2 + bx + c$ . $\newline$ $a = 2$ , $b = 5$ , $c = 3$
Find Multiplying Numbers: Find two numbers that multiply to $a*c$ ( $2*3 = 6$ ) and add up to $b$ ( $5$ ). $\newline$ The numbers that satisfy these conditions are $2$ and $3$ because $2*3 = 6$ and $2+3 = 5$ .
Rewrite Middle Term: Rewrite the middle term $5t$ using the two numbers found in the previous step. $2t^2 + 5t + 3$ can be rewritten as $2t^2 + 2t + 3t + 3$ .
Factor by Grouping: Factor by grouping. Group the first two terms and the last two terms. $\newline$ $(2t^2 + 2t) + (3t + 3)$
Factor out Common Factor: Factor out the greatest common factor from each group. $2t(t + 1) + 3(t + 1)$
Check Factored Form: Since both groups contain the common factor $(t + 1)$ , factor it out. $(2t + 3)(t + 1)$
Check Factored Form: Since both groups contain the common factor $(t + 1)$ , factor it out. $(2t + 3)(t + 1)$ Check the factored form by expanding it to ensure it equals the original expression. $(2t + 3)(t + 1) = 2t^2 + 2t + 3t + 3 = 2t^2 + 5t + 3$