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Factor. $\newline$ $y^2 + 10y + 21$

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Factor quadratics with leading coefficient 1

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Q. Factor.

\newline

y^2 + 10y + 21

Identify Variables: Identify $a$ , $b$ , and $c$ in the quadratic expression $y^2 + 10y + 21$ . Compare $y^2 + 10y + 21$ with the standard quadratic form $ax^2 + bx + c$ . Here, $a = 1$ , $b = 10$ , and $c = 21$ .
Find Product and Sum: Find two numbers that multiply to $a*c$ (which is $21$ ) and add up to $b$ (which is $10$ ). $\newline$ We need to find two numbers whose product is $21$ and whose sum is $10$ . $\newline$ The numbers $3$ and $7$ satisfy these conditions because: $\newline$ $3 \times 7 = 21$ and $3 + 7 = 10$ .
Split Middle Term: Write the quadratic expression using the two numbers found in Step $2$ to split the middle term. $\newline$ $y^2 + 10y + 21$ can be rewritten as $y^2 + 3y + 7y + 21$ .
Factor by Grouping: Factor by grouping. $\newline$ Group the terms to factor by common terms: $\newline$ $(y^2 + 3y) + (7y + 21)$ $\newline$ Factor out the common factors from each group: $\newline$ $y(y + 3) + 7(y + 3)$
Factor out Binomial: Factor out the common binomial factor. $\newline$ Since both terms contain $(y + 3)$ , factor it out: $\newline$ $(y + 3)(y + 7)$

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