Determine whether the statement is true or false. If it is...

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false.
If $b^{2}-4 a c<0,$ then $a x^{2}+b x+c=0(a \neq 0)$ has no real roots.

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Key Concepts

Real Roots Analysis

The analysis of real roots in a quadratic equation involves evaluating the discriminant. Specifically, if the discriminant is negative, it implies that the equation does not have any real roots since the solutions, as obtained from the quadratic formula, involve taking the square root of a negative number, resulting in complex solutions.

Quadratic Equation

A quadratic equation is an algebraic equation of the second degree, typically written in the form ax² + bx + c = 0, where a, b, and c are constants and a is not equal to zero. This form is fundamental in algebra and leads to various methods of solving, including factoring, completing the square, and using the quadratic formula.

Discriminant

The discriminant in a quadratic equation is the expression b² - 4ac, which plays a crucial role in determining the nature and number of roots of the equation. Its value indicates whether the quadratic will have two real and distinct roots (if positive), one real repeated root (if zero), or two complex conjugate roots (if negative).

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