Question

consider the equation f(x) = x^{2} +4x+7 how many and what type of roots does the equation f(x) = 0 have

Answer 1

We can determine the roots of the quadratic equation f(x) = x^2 + 4x + 7 by using the quadratic formula, which states that for a quadratic equation of the form ax^2 + bx + c = 0, the roots are given by:

x = (-b ± sqrt(b^2 - 4ac)) / 2a

In this case, a = 1, b = 4, and c = 7. Substituting these values into the quadratic formula, we get:

x = (-4 ± sqrt(4^2 - 4(1)(7))) / 2(1)

Simplifying this expression, we get:

x = (-4 ± sqrt(16 - 28)) / 2

x = (-4 ± sqrt(-12)) / 2

Since the square root of a negative number is not a real number, the roots of the equation f(x) = x^2 + 4x + 7 are complex conjugates. Therefore, the equation has no real roots.