Algebra • Step-by-Step

Solve x² - 1 = 0 with Steps

Find the roots of x² - 1 = 0 step by step using difference of squares (x - 1)(x + 1) = 0 yielding x = ±1.

Problem Statement

x^2 - 1 = 0

Final Result

x_1 = -1, \quad x_2 = 1

Formula Required: Difference of Squares

Take square roots: x² = 1 ⇒ x = ±1.

x^2 - 1 = (x - 1)(x + 1) = 0

6 Steps

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

x^2 - 1 = 0

Find two numbers that multiply to $c = -1$ and add up to $b = 0$.

(1) \times (-1) = -1, \quad (1) + (-1) = 0

Write the quadratic expression in factored form.

(x + 1)(x - 1) = 0

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

x + 1 = 0 \implies x_1 = -1 \quad \text{or} \quad x - 1 = 0 \implies x_2 = 1

Verify each root by substituting back into the original equation.

\text{Check } x = -1: \; (-1)^2 - 1 = 1 + 0 - 1 = 0 \;\checkmark \\[6pt] \text{Check } x = 1: \; (1)^2 - 1 = 1 + 0 - 1 = 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} = 0, \quad \text{Product: } x_1 \cdot x_2 = \frac{c}{a} = -1

Interactive Calculator

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

x = 1 and x = -1.