Algebra • Step-by-Step

Solve x² - 7x + 12 = 0 with Steps

Find the roots of x² - 7x + 12 = 0 step by step. Factor into (x - 3)(x - 4) = 0 to obtain x = 3 and x = 4.

Problem Statement

x^2 - 7x + 12 = 0

Final Result

x_1 = 3, \quad x_2 = 4

Formula Required: Factoring Trinomials

Find numbers p and q such that p + q = 7 and p · q = 12 (3 and 4).

x^2 - (p+q)x + pq = (x - p)(x - q) = 0

6 Steps

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

x^2 - 7x + 12 = 0

Find two numbers that multiply to $c = 12$ and add up to $b = -7$.

(-3) \times (-4) = 12, \quad (-3) + (-4) = -7

Write the quadratic expression in factored form.

(x - 3)(x - 4) = 0

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

x - 3 = 0 \implies x_1 = 3 \quad \text{or} \quad x - 4 = 0 \implies x_2 = 4

Verify each root by substituting back into the original equation.

\text{Check } x = 3: \; (3)^2 - 7(3) + 12 = 9 - 21 + 12 = 0 \;\checkmark \\[6pt] \text{Check } x = 4: \; (4)^2 - 7(4) + 12 = 16 - 28 + 12 = 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} = 7, \quad \text{Product: } x_1 \cdot x_2 = \frac{c}{a} = 12

Interactive Calculator

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

x = 3 and x = 4.