Algebra • Step-by-Step

Solve x² - 5x + 6 = 0 with Steps

Solve x² - 5x + 6 = 0 step by step using factoring and the quadratic formula. Find the roots x = 2 and x = 3 with full working.

Problem Statement

x^2 - 5x + 6 = 0

Final Result

x_1 = 2, \quad x_2 = 3

Formula Required: Quadratic Formula & Factoring

Identify a = 1, b = -5, c = 6. Find two numbers that multiply to 6 and add to -5 (-2 and -3).

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \quad \text{or} \quad (x - p)(x - q) = 0

6 Steps

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

x^2 - 5x + 6 = 0

Find two numbers that multiply to $c = 6$ and add up to $b = -5$.

(-2) \times (-3) = 6, \quad (-2) + (-3) = -5

Write the quadratic expression in factored form.

(x - 2)(x - 3) = 0

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

x - 2 = 0 \implies x_1 = 2 \quad \text{or} \quad x - 3 = 0 \implies x_2 = 3

Verify each root by substituting back into the original equation.

\text{Check } x = 2: \; (2)^2 - 5(2) + 6 = 4 - 10 + 6 = 0 \;\checkmark \\[6pt] \text{Check } x = 3: \; (3)^2 - 5(3) + 6 = 9 - 15 + 6 = 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} = 5, \quad \text{Product: } x_1 \cdot x_2 = \frac{c}{a} = 6

Interactive Calculator

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

The roots are x = 2 and x = 3. By factoring: (x - 2)(x - 3) = 0.