1Cademy - Factoring $$18x^2

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Factoring Trinomials of the Form $ax^2 + bx + c$ Using Trial and Error

Example

Factoring $18x^2 - 37xy + 15y^2$

To factor $18x^2 - 37xy + 15y^2$ using trial and error, first ensure the trinomial is in descending order. Then, determine the factor pairs for the leading term $18x^2$ , which are $x \cdot 18x$ , $2x \cdot 9x$ , and $3x \cdot 6x$ . Next, identify the factor pairs for the last term $15y^2$ . Because the last term is positive and the middle term coefficient ( $-37$ ) is negative, use negative factors: $-y \cdot -15y$ and $-3y \cdot -5y$ . Systematically test combinations of these first and last term factors to find the pair whose inner and outer products sum to $-37xy$ . As shown in the combinations table, testing the binomials $(2x - 3y)$ and $(9x - 5y)$ yields an inner product of $-27xy$ and an outer product of $-10xy$ , which sum to exactly $-37xy$ . Therefore, the correct factorization is $(2x - 3y)(9x - 5y)$ . Finally, verify the result by multiplying: $(2x - 3y)(9x - 5y) = 18x^2 - 10xy - 27xy + 15y^2 = 18x^2 - 37xy + 15y^2$ .

Updated 2026-04-29

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OpenStax Intermediate Algebra

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