1Cademy - Solving $$3p^2 + 2p + 9 = 0$$ Using the Quadratic Formula

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How to Solve a Quadratic Equation Using the Quadratic Formula

Example

Solving $3p^2 + 2p + 9 = 0$ Using the Quadratic Formula

Solve $3p^2 + 2p + 9 = 0$ by applying the Quadratic Formula. This example demonstrates the outcome when the expression under the square root evaluates to a negative number, resulting in no real solution.

Step 1 — Identify $a$ , $b$ , $c$ . The equation is already in standard form $ax^2 + bx + c = 0$ . Here $a = 3$ , $b = 2$ , and $c = 9$ .

Step 2 — Substitute into the Quadratic Formula:

$p = \frac{-2 \pm \sqrt{2^2 - 4(3)(9)}}{2(3)}$

Step 3 — Simplify. Compute inside the square root: $2^2 = 4$ and $4(3)(9) = 108$ , so $4 - 108 = -104$ :

$p = \frac{-2 \pm \sqrt{-104}}{6}$

Since $\sqrt{-104}$ is not a real number — the square root of a negative number does not exist among the real numbers — there is no real solution.

When all three coefficients $a$ , $b$ , and $c$ are positive and the equation equals zero, the expression $b^2 - 4ac$ often turns out to be negative because the product $4ac$ exceeds $b^2$ . In such cases, the Quadratic Formula reveals that no real number can satisfy the equation.

Updated 2026-04-21

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OpenStax Elementary Algebra

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