(xxi) Solve the quadratic equation:
$$2m^2 + 19m + 30 = 0$$
Solution:
We are given the equation:
$$2m^2 + 19m + 30 = 0$$We need to find two numbers that multiply to (2 × 30 = 60) and add up to 19. These numbers are 15 and 4. We split the middle term:
$$2m^2 + 15m + 4m + 30 = 0$$Now, we factor by grouping:
$$m(2m + 15) + 2(2m + 15) = 0$$Factor out the common term (2m + 15):
$$(2m + 15)(m + 2) = 0$$By the zero-product property, either the first factor is zero or the second factor is zero:
$$2m + 15 = 0 \quad \text{or} \quad m + 2 = 0$$Solving for m in each case:
$$2m = -15 \quad \text{or} \quad m = -2$$This gives us the final solutions:
$$m = -\frac{15}{2} \quad \text{or} \quad m = -2$$