Algebra • Step-by-Step

Solve x² + 6x + 9 = 0 with Steps

Solve x² + 6x + 9 = 0 step by step. Recognize (x + 3)² = 0 with discriminant Δ = 0 giving single root x = -3.

Problem Statement

x^2 + 6x + 9 = 0

Final Result

x = -3

Formula Required: Perfect Square Trinomial

b² - 4ac = 36 - 36 = 0, meaning there is one repeated root.

(x + 3)^2 = 0 \implies x = -3

6 Steps

Write the equation in standard quadratic form $ax^2 + bx + c = 0$.

x^2 + 6x + 9 = 0

Find two numbers that multiply to $c = 9$ and add up to $b = 6$.

(3) \times (3) = 9, \quad (3) + (3) = 6

Write the quadratic expression in factored form (perfect square trinomial).

(x + 3)^2 = 0

Set the factor equal to zero and solve for $x$.

x + 3 = 0 \implies x = -3

Verify the solution by substituting back into the original equation.

\text{Check } x = -3: \; (-3)^2 + 6(-3) + 9 = 9 - 18 + 9 = 0 \;\checkmark

Structural Insight (Vieta's Formulas): Notice how the sum and product of the roots directly confirm the original coefficients.

\text{Sum of roots: } x_1 + x_2 = -\frac{b}{a} = -6, \quad \text{Product: } x_1 \cdot x_2 = \frac{c}{a} = 9

Interactive Calculator

Enter any equation like 2x + 8 = 20 or x² - 5x + 6 = 0 to get step-by-step solutions.

x = -3 (repeated root with multiplicity 2).