Question:

Step 1: Rearrange the given equation to standard form. We start with the given condition: $$a^2 + b^2 + c^2 + 4 = ab + bc + 2c + 2a$$ Moving all terms to one side: $$a^2 + b^2 + c^2 + 4 - ab - bc - 2c - 2a = 0$$ Step 2: Multiply the equation by 2 to facilitate grouping into perfect squares. Multiplying both sides by 2: $$2a^2 + 2b^2 + 2c^2 + 8 - 2ab - 2bc - 4c - 4a = 0$$ Step 3: Regroup the terms to express them as sums of perfect squares. We strategically rearrange the terms: $$2a^2 + 2b^2 + 2c^2 + 8 - 2ab - 2bc - 4c - 4a$$ $$= (a^2 - 2ab + b^2) + (b^2 - 2bc + c^2) + (a^2 - 4a + 4) + (c^2 - 4c + 4)$$ Step 4: Express each group as a perfect square. Recognizing the perfect square patterns: $$(a - b)^2 + (b - c)^2 + (a - 2)^2 + (c - 2)^2 = 0$$ Step 5: Use the property that a sum of non-negative terms equals zero. Since each squared term is non-negative (being a perfect square), and their sum equals zero, each term must individually equal zero: $$(a - b)^2 = 0 \implies a = b$$ $$(b - c)^2 = 0 \implies b = c$$ $$(a - 2)^2 = 0 \implies a = 2$$ $$(c - 2)^2 = 0 \implies c = 2$$ Therefore: $a = b = c = 2$ Step 6: Form the quadratic equation with the found values. Substituting $a = 2$, $b = 2$, and $c = 2$ into $ax^2 + bx + c = 0$: $$2x^2 + 2x + 2 = 0$$ Dividing by 2: $$x^2 + x + 1 = 0$$ Step 7: Calculate the discriminant to determine the nature of roots. For the quadratic $x^2 + x + 1 = 0$, the discriminant is: $$\Delta = b^2 - 4ac = (1)^2 - 4(1)(1) = 1 - 4 = -3$$ Since $\Delta = -3 < 0$, the roots are complex (imaginary). **Final Answer:** The roots of $ax^2 + bx + c = 0$ are **imaginary**. The correct option is **Option 3: imaginary**.