To factorise the quadratic equation 6x² + 5x + 6, we need to look for two numbers that multiply to the product of the coefficient of x² (which is 6) and the constant term (which is 6) and also add up to the coefficient of x (which is 5).
First, we calculate the product: 6 * 6 = 36. Now, we need two numbers that multiply to 36 and add up to 5. Unfortunately, there are no such integer pairs that meet these criteria. This indicates that the quadratic does not factor neatly with integer coefficients.
An alternative method is to use the quadratic formula, x = (-b ± √(b² – 4ac)) / 2a, where a = 6, b = 5, and c = 6:
- Calculate the discriminant: b² – 4ac = 5² – 4 * 6 * 6 = 25 – 144 = -119.
- Since the discriminant is negative, this tells us that the roots of the equation are complex numbers, meaning we cannot factor it over the real numbers.
In conclusion, the equation 6x² + 5x + 6 cannot be factored into simpler expressions with real coefficients, as it has complex roots. It is already in its simplest form.