To factorise the quadratic expression x² + 2x + 8, we first look for two numbers that multiply to the constant term (8) and add up to the coefficient of the linear term (2). Unfortunately, there are no two real numbers that satisfy these conditions.
When we try to find two numbers that multiply to 8, we come up with pairs like (1, 8) or (2, 4), but none of these pairs add up to 2. Therefore, this expression cannot be factored into linear terms using real numbers.
As a result, the quadratic remains in its original form or can be expressed using the quadratic formula:
x = (-b ± √(b² – 4ac)) / 2a
In our case, a = 1, b = 2, and c = 8. Plugging these into the formula provides the roots of the equation rather than a factorization:
x = (-2 ± √(2² – 4 * 1 * 8)) / (2 * 1) = (-2 ± √(4 – 32)) / 2 = (-2 ± √-28) / 2
= (-2 ± 2i√7) / 2 = -1 ± i√7
Thus, the quadratic does not factor neatly with real numbers, and the expression retains its form.