State the number of complex roots of each equation then find the roots.
I cant figure out these two!
1. 3x - 5 =0
2. c^2 + 2c + 1 = 0
5/3 is a single real number
(c+1)(c+1) = 0
c = -1 or -1
two identical real roots (vertex of parabola on axis)
To determine the number of complex roots of an equation, we need to consider its discriminant. The discriminant is calculated differently depending on the type of equation.
1. 3x - 5 = 0:
This equation is a linear equation of the form ax + b = 0. It does not have complex roots because it is a linear equation, so there is only one solution. To solve it, we isolate x:
3x - 5 = 0
3x = 5
x = 5/3
Therefore, the equation 3x - 5 = 0 has one real root, x = 5/3.
2. c^2 + 2c + 1 = 0:
This equation is a quadratic equation of the form ax^2 + bx + c = 0. To determine the number of complex roots, we need to calculate the discriminant using the formula Δ = b^2 - 4ac.
In this case, a = 1, b = 2, and c = 1. Substituting these values into the formula:
Δ = (2^2) - 4(1)(1)
Δ = 4 - 4
Δ = 0
The discriminant Δ is equal to 0. When the discriminant is 0, it means the quadratic has two identical real roots. To find the roots, we can use the quadratic formula:
x = (-b ± √Δ) / (2a)
Substituting the values into the formula:
x = (-2 ± √0) / (2(1))
x = (-2 ± 0) / 2
x = -2 / 2
x = -1
Therefore, the equation c^2 + 2c + 1 = 0 has two real roots, both equal to -1.