To determine the type of roots for the given quadratic equation, we will use the discriminant, which is part of the quadratic formula. The quadratic formula is given by:
x = (-b ± √(b² – 4ac)) / 2a
In our equation, 16x² + 182x + 49 = 0, we can identify the coefficients as follows:
- a = 16
- b = 182
- c = 49
The discriminant (D) is calculated using the formula:
D = b² – 4ac
Now, let’s compute the discriminant:
D = (182)² – 4(16)(49)
Calculating (182)²:
182 × 182 = 33124
Calculating 4(16)(49):
4 × 16 × 49 = 3136
Now, substituting back into the discriminant:
D = 33124 – 3136
D = 30088
Since the discriminant is greater than zero (D > 0), this tells us that the equation has two distinct real roots. In summary:
The equation 16x² + 182x + 49 = 0 has two distinct real roots.