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If a quadratic equation is prime, that means it can not be factored. The word 'prime' itself means that the particular expression or number can not be factored. A prime quadratic equation does not have rational roots or x intercepts. That is because a prime quadratic equation can not be written in the form a(x - p)(x - q) = 0. Hence, in order to solve a prime quadratic equation, you have to apply the quadratic formula.

For example, the quadratic equation x2 + 2x + 2 = 0 is a prime quadratic equation. Thus it can not be factored by the box method or by splitting its middle term. In order to solve this prime quadratic equation, you have to apply the quadratic formula, as follows:

Comparing x2 + 2x + 2 = 0 with standard quadratic equation ax2 + bx + c = 0, you get
• a = 1
• b = 2
• c = 2
x = {-b +- sqrt(b^2 - 4ac)}/{2a}
x = {-2 +- sqrt{2^2 - 4*1*2}}/{2*1}
either x = -i - 1 or x = i - 1
The quadratic equation x2 + 2x + 2 = 0 does not have rational roots as it is a prime quadratic equation. It is solved by the quadratic formula to obtain its two roots x = -i - 1 or x = i - 1