Algebra • Step-by-Step

Solve x² + 1 = 0 with Steps

Q: Are the roots of x² + 1 = 0 real or complex?

They are complex (pure imaginary): x = i and x = -i.

Find the complex roots of x² + 1 = 0 step by step. Shows discriminant Δ = -4 yielding conjugate imaginary roots x = ±i.

Problem Statement

x^2 + 1 = 0

Final Result

x = \pm i

Formula Required: Complex Roots & Imaginary Unit

When x² is negative, solutions are imaginary numbers involving i.

x^2 = -1 \implies x = \pm \sqrt{-1} = \pm i

Step-by-Step Solution

8 Steps

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

x^2 + 1 = 0

Identify the coefficients $a$, $b$, and $c$.

a = 1, \quad b = 0, \quad c = 1

Calculate the discriminant $\Delta = b^2 - 4ac$ (tells us how many real roots exist). Since $\Delta < 0$ is negative, there are no real roots; the equation has two complex conjugate roots.

\Delta = (0)^2 - 4(1)(1) = 0 - (4) = -4

Apply the quadratic formula.

x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{0 \pm \sqrt{-4}}{2}

Rewrite the negative square root using the imaginary unit $i = \sqrt{-1}$:

x = \frac{0 \pm \sqrt{4}\;i}{2}

Simplify the two complex conjugate roots:

x = 0 \pm \,i

Verification for complex conjugate pair:

f(0 \pm i) = (1)(0 \pm i)^2 + (0)(0 \pm i) + (1) = 0 \;\checkmark

Structural Insight (Vieta's Formulas): The roots directly mirror the coefficients of the polynomial.

\text{Sum of roots: } x_1 + x_2 = -\frac{b}{a} = 0, \quad \text{Product of roots: } x_1 \cdot x_2 = \frac{c}{a} = 1

Try It Yourself or Edit Equation

Interactive Calculator

Awaiting Input

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

Common Questions About Solve x² + 1 = 0

Are the roots of x² + 1 = 0 real or complex?