Completing the Square to Find Complex Roots

Consider the quadratic equation $x^2+2x=3$.

Jayden solves the equation by completing the square: $(x+1)^2 = 4$, so $x+1 = \pm\sqrt4$. "That means $x = -1\pm\sqrt4$," he says.

Mia solves the equation by subtracting $3$ from both sides and factoring the left hand side: $x^2+2x-3 = 0$, so $(x+3)(x-1) = 0$. "By the zero product property, that means $x=-3$ or $x=1$," she says.

A classroom argument soon follows.

• Kevin says Mia is right, but Jayden is wrong because his answer has a square root in it while Mia's doesn't.
• Jordan says Jayden is right, but Mia is wrong because her answer doesn't have square roots in it.
• Jacques says both Mia and Jayden are right, it's just that their answers are expressed in different ways.