Algebra • Step-by-Step

Solve x² + 4 = 0 with Steps

Step-by-step solution for the quadratic equation x² + 4 = 0. Shows how taking the square root of negative numbers yields imaginary roots x = ±2i.

Problem Statement

x^2 + 4 = 0

Final Result

x = \pm 2\,i

Formula Required: Imaginary Numbers Formula

√(-4) = √(4) · √(-1) = 2i.

x^2 = -4 \implies x = \pm \sqrt{-4} = \pm 2i

Step-by-Step Solution

8 Steps

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

x^2 + 4 = 0

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

a = 1, \quad b = 0, \quad c = 4

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)(4) = 0 - (16) = -16

Apply the quadratic formula.

x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{0 \pm \sqrt{-16}}{2}

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

x = \frac{0 \pm \sqrt{16}\;i}{2}

Simplify the two complex conjugate roots:

x = 0 \pm 2\,i

Verification for complex conjugate pair:

f(0 \pm 2i) = (1)(0 \pm 2i)^2 + (0)(0 \pm 2i) + (4) = 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} = 4

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Common Questions About Solve x² + 4 = 0

What are the roots of x² + 4 = 0?

x = 2i and x = -2i.