To find the factors of the quadratic expression 2x² + 3x + 54, we will use the technique of factoring by grouping or the quadratic formula if necessary.
First, we can identify the coefficients from the given expression:
- a: 2 (the coefficient of x²)
- b: 3 (the coefficient of x)
- c: 54 (the constant term)
Next, we can calculate the discriminant (D) of the quadratic equation using the formula:
D = b² – 4ac
In this case:
- D = (3)² – 4(2)(54)
- D = 9 – 432
- D = -423
Since the discriminant is negative (D < 0), this indicates that the quadratic expression does not have real roots and hence cannot be factored into linear factors with real coefficients.
This means that 2x² + 3x + 54 is either prime or requires complex numbers for factoring. Therefore, express it as:
(2x + 9 + i√423/2)(2x + 9 – i√423/2)
In conclusion, the expression 2x² + 3x + 54 does not have real factorable components and can only be expressed in complex terms.