1Cademy - Solving $$x(x + 6) + 4 = 0$$ Using the Quadratic Formula

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How to Solve a Quadratic Equation Using the Quadratic Formula

Example

Solving $x(x + 6) + 4 = 0$ Using the Quadratic Formula

Solve $x(x + 6) + 4 = 0$ by applying the Quadratic Formula. This example demonstrates a case where the equation must first be rewritten in standard form by distributing before the formula can be used.

Distribute to get standard form. Apply the Distributive Property to expand $x(x + 6)$ :

$x^2 + 6x + 4 = 0$

Step 1 — Identify $a$ , $b$ , $c$ . The equation is now in standard form. Here $a = 1$ , $b = 6$ , and $c = 4$ .

Step 2 — Substitute into the Quadratic Formula:

$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(4)}}{2(1)}$

Step 3 — Simplify. Compute inside the square root: $6^2 = 36$ and $4(1)(4) = 16$ , so $36 - 16 = 20$ :

$x = \frac{-6 \pm \sqrt{20}}{2}$

Simplify the radical. The largest perfect square factor of $20$ is $4$ : $\sqrt{20} = \sqrt{4 \cdot 5} = 2\sqrt{5}$ :

$x = \frac{-6 \pm 2\sqrt{5}}{2}$

Factor out the common factor of $2$ in the numerator: $\frac{2(-3 \pm \sqrt{5})}{2}$ . Cancel the common factor:

$x = -3 \pm \sqrt{5}$

Rewrite as two solutions:

$x = -3 + \sqrt{5} \qquad \text{or} \qquad x = -3 - \sqrt{5}$

When the equation is not already in standard form — for instance, when a product like $x(x + 6)$ needs to be expanded — the Distributive Property must be applied as a preliminary step before identifying $a$ , $b$ , and $c$ . After simplification, if a common factor appears in all terms of the numerator and in the denominator, it should be factored out and canceled to produce the simplest form of the solution.

Updated 2026-04-21

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OpenStax Elementary Algebra

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