1Cademy - Solving $$2y^2 = 13y + 45$$ by Factoring

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Quadratic Equation

Example

Solving $2y^2 = 13y + 45$ by Factoring

Solve $2y^2 = 13y + 45$ by first rearranging into standard form and then factoring.

Step 1 — Write in standard form. The equation is not yet in the form $ay^2 + by + c = 0$ because $13y + 45$ appears on the right side. Subtract $13y$ and $45$ from both sides:

$2y^2 - 13y - 45 = 0$

Step 2 — Factor the quadratic expression. The trinomial $2y^2 - 13y - 45$ has a leading coefficient of $2$ , so use a factoring method for trinomials of the form $ay^2 + by + c$ :

$(2y + 5)(y - 9) = 0$

Step 3 — Apply the Zero Product Property. Set each factor equal to zero:

$2y + 5 = 0 \quad \text{or} \quad y - 9 = 0$

Step 4 — Solve each linear equation:

$y = -\frac{5}{2} \quad \text{or} \quad y = 9$

Step 5 — Check both solutions by substituting into the original equation $2y^2 = 13y + 45$ :

For $y = -\frac{5}{2}$ : $2\left(-\frac{5}{2}\right)^2 = 2 \cdot \frac{25}{4} = \frac{25}{2}$ and $13\left(-\frac{5}{2}\right) + 45 = -\frac{65}{2} + \frac{90}{2} = \frac{25}{2}$ ✓

For $y = 9$ : $2(9)^2 = 162$ and $13(9) + 45 = 117 + 45 = 162$ ✓

The solutions are $y = -\frac{5}{2}$ and $y = 9$ . This example highlights the importance of Step 1: because the original equation has nonzero terms on both sides, it must first be rewritten in standard form before the expression can be factored and the Zero Product Property applied.

Updated 2026-04-30

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