Solve $x^2 + 4x + 5 = 0$.

How we read your problem

A quadratic equation in x: solve with the quadratic formula.

Step-by-step solution

1. 1

   Start with the equation

   $${ x}^{2}+4~ x+5 = 0$$
2. 2

   Move every term to one side to get standard form

   $$x^{2} + 4x + 5 = 0$$
3. 3

   Compute the discriminant

   $$\Delta = b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot 5 = -4$$
4. 4

   Δ < 0, so there are no real roots; the two roots are complex conjugates

5. 5

   Apply the quadratic formula with √(−1) = i

   $$x = \frac{-4 \pm \sqrt{-4}}{2} = \frac{-4 \pm 2\,i}{2}$$
6. 6

   Simplify

   $$x = -2 \pm i$$

Formulas used

• Quadratic formula

  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Discriminant

  $$\Delta = b^2 - 4ac$$

• Vertex

  $$\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)$$

The discriminant is negative, so there are no real solutions; the solutions are a pair of complex conjugates.

Verification

• Vieta: sum of roots = -4 = −b/a = -4; product = 5 = c/a = 5