Question

Under what conditions would the well-known quadratic formula $$ \text { Roots }=\frac{-b \pm \sqrt{b^{2}-4 a c}}{2 a} $$ not be effectively computable? (Assume that you are working with real numbers.)

Short answer

The formula is not computable if \(b^2 - 4ac < 0\) or \(a = 0\).

Step-by-step solution

Understand the Discriminant

In the quadratic formula, the discriminant is the expression under the square root: \(b^2 - 4ac\). The discriminant determines the nature of the roots of the quadratic equation. If the discriminant is negative, the roots are not real numbers.

Analyze Conditions for Real Roots

For the roots to be real, the discriminant must be non-negative, meaning \(b^2 - 4ac \geq 0\). This is because the square root of a negative number is not defined in the real number system.

Explore Issues with Division by Zero

The quadratic formula also involves division by \(2a\). If \(a = 0\), the expression \(2a\) becomes zero, leading to division by zero, which is undefined. Therefore, \(a\) must be non-zero for the formula to be computable.

Summary of Non-Computable Conditions

The quadratic formula is not effectively computable when the discriminant is negative (\(b^2 - 4ac < 0\)) or when \(a = 0\). In either case, the formula cannot provide a valid solution within the real number system.

Key concepts

Discriminant

The discriminant is a crucial element in understanding the nature of roots for a quadratic equation. In the quadratic formula \( \text{Roots} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), the discriminant is represented by the expression \( b^2 - 4ac \) found under the square root sign. This element tells us a lot about the potential roots:

• If \( b^2 - 4ac > 0 \), then the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), we have exactly one real root, meaning the roots are equal.
• If \( b^2 - 4ac < 0 \), the equation has no real roots, making them complex. This is crucial as it fixes the conditions under which the equation is non-computable in the realm of real numbers.

So, the discriminant indicates whether the solutions can actually exist within the real number system, highlighting its importance in determining root realness.

Real Roots

To determine if a quadratic equation has real roots, the discriminant must be non-negative. As mentioned earlier, the discriminant is \( b^2 - 4ac \). For real roots to exist in a quadratic equation, this value must meet the condition \( b^2 - 4ac \geq 0 \). This ensures that the square root operation in the formula works smoothly without intruding into the realm of imaginary numbers.
When the discriminant is positive, the result is two distinct real numbers, meaning the parabola intersects the x-axis at two points. When it equals zero, the result is a single real number, indicating the parabola touches the x-axis only once. Thus, the concept of real roots is essentially tied to the sign of the discriminant. This understanding helps in knowing whether the quadratic solutions can be appropriately plotted and understood within the scope of real arithmetic.

Division by Zero

Division by zero is a mathematical operation that is undefined and can cause problems in calculations. Within the quadratic formula \( \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), there is a potential for division by zero if \( a = 0 \). This would make the term \( 2a \) equal to zero, rendering the division impossible.
It's important to understand that when \( a = 0 \), the equation ceases to be quadratic. Instead, it becomes linear, which significantly alters the manner in which roots are calculated. Hence, for the quadratic formula to remain valid, \( a \) should never equal zero. This small but crucial condition ensures that the formula remains effective and computable, avoiding the undefined nature of dividing by zero, which would otherwise disrupt the process of finding roots.

Computability

Computability in the context of the quadratic formula refers to whether you can find real and meaningful solutions using this method. Several factors can affect this:

• If the discriminant is negative, the roots are non-real, moving into the domain of complex numbers and thus out of reach for those seeking real solutions.
• If \( a = 0 \), the formula leads to division by zero, an operation that is undefined in mathematics.

Both these conditions render the quadratic formula ineffective in providing real results. Ensuring that the discriminant is non-negative and that \( a eq 0 \) are the necessary checks to keep the formula computable within the real number system. By understanding these conditions, you avoid errors and pitfalls, thus securing the integrity of your calculations.