1Cademy - Factoring $$10y^2

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

Example

Factoring $10y^2 - 37y + 7$

To factor the trinomial $10y^2 - 37y + 7$ completely using trial and error:

Step 1 — Verify the expression is in descending order. The trinomial $10y^2 - 37y + 7$ is arranged correctly.

Step 2 — Find the factors of the first term, $10y^2$ . The possible binomial first terms are $y \cdot 10y$ and $2y \cdot 5y$ .

Step 3 — Find the factors of the last term, $7$ . Since the last term is positive and the middle term is negative, both factors must be negative: $-1$ and $-7$ .

Step 4 — Test combinations to match the middle term of $-37y$ .

For $y$ $y$ and $10y$ $10 y$ :
- $(y - 1)(10y - 7)$ gives a middle term of $-17y$ .
- $(y - 7)(10y - 1)$ gives a middle term of $-71y$ .
For $2y$ $2 y$ and $5y$ $5 y$ :
- $(2y - 1)(5y - 7)$ gives a middle term of $-19y$ .
- $(2y - 7)(5y - 1)$ gives a middle term of $-37y$ . ✓

The correct factors are $(2y - 7)(5y - 1)$ .

Step 5 — Check by multiplying: $(2y - 7)(5y - 1) = 10y^2 - 2y - 35y + 7 = 10y^2 - 37y + 7$ .

The completely factored form is $(2y - 7)(5y - 1)$ .

Updated 2026-06-03

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OpenStax Intermediate Algebra

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