Number of real roots of the quadratic equation 3x 2 + 4x + 25 = 0

SOLUTION

Concept:

Discriminant for a quadratic equation:

For a quadratic equation \(ax^2+bx+c= 0\) where \(a\ne 0\) and b and c are real numbers the discriminant of the quadratic equation is given by:

\(Δ = b^2-4ac\)

Nature of roots of a quadratic equation:

Consider a quadratic equation \(ax^2+bx+c= 0\) where \(a\ne 0\) and b and c are real numbers the nature of the roots is determined as follows:

1. If \(Δ >0\) then the roots are real numbers. If \(Δ\) is a perfect square then the roots are rational otherwise they are irrational.
2. If \(Δ = 0\) then the roots are real and equal.
3. If \(Δ < 0\) then the roots are imaginary; in fact they are complex conjugates of each other.

Calculation:

Given quadratic equation: 3x2 + 4x + 25 = 0

 where \(a\ne 0\) and b and c are real numbers.

\(Δ = b^2-4ac\)

Δ = 16 - 4 x 3 x 25 = -286

Δ < 0

Since Δ  not greater than zero.

∴ The given quadratic equation has no real and distinct roots.