Write Whether The Following Statements Are True Or False Justify Your Answers Every Quadratic Equation Has At Least One Real Root

When it comes to quadratic equations, there’s often a sense of intrigue and wonder about their properties. Quadratic equations are an essential part of mathematics and find applications in various fields. One common question that arises is whether every quadratic equation has at least one real root. In this article, we’ll delve into this mathematical inquiry, explore the nature of quadratic equations, and provide a definitive answer to the question at hand.

Understanding Quadratic Equations

Before we dive into the question of real roots, let’s make sure we’re on the same page about what quadratic equations are. A quadratic equation is a polynomial equation of the second degree, represented in the form:

��2+��+�=0ax2+bx+c=0

Here, $a$, $b$, and $c$ are coefficients, and $x$ is the variable we’re solving for. Quadratic equations can take on various forms, but they all share this fundamental structure.

The Discriminant: Key to Real Roots

To determine whether a quadratic equation has real roots, we need to examine a crucial parameter called the discriminant ($D$). The discriminant is calculated using the formula:

�=�2−4��D=b2−4ac

The value of the discriminant ($D$) plays a significant role in revealing the nature of the roots of the quadratic equation. Here’s how it works:

1. If $D>0$: When the discriminant is greater than zero, the quadratic equation has two distinct real roots. In this case, the graph of the quadratic equation intersects the x-axis at two different points.

2. If $D=0$: When the discriminant is equal to zero, the quadratic equation has one real root, which is repeated. Geometrically, this means that the graph of the equation touches the x-axis at a single point.

3. If $D<0$: When the discriminant is less than zero, the quadratic equation has no real roots. In this scenario, the graph of the equation does not intersect the x-axis at any point.

Justifying the Statements

Now, let’s address the original question: “Does every quadratic equation have at least one real root?” To answer this, we can break it down based on the cases we discussed above:

• If the discriminant $D$ is greater than zero, the quadratic equation has two real roots. So, the statement is true in this case.

• If the discriminant $D$ is equal to zero, the quadratic equation has one real root (which is repeated). Thus, the statement is true in this scenario as well.

• If the discriminant $D$ is less than zero, the quadratic equation has no real roots. In this case, the statement is false because there are quadratic equations without real roots.

So, the answer to the question depends on the value of the discriminant $D$. Every quadratic equation does have at least one real root if the discriminant is greater than or equal to zero. However, if the discriminant is negative, the statement is false as there are cases where quadratic equations have no real roots.

Frequently Asked Questions (FAQs)

1. What is the discriminant of a quadratic equation?

The discriminant of a quadratic equation is a mathematical parameter represented by the symbol $D$. It is calculated using the formula �=�2−4��D=b2−4ac and is used to determine the nature of the roots of the equation.

2. Can a quadratic equation have more than two real roots?

No, a quadratic equation can have at most two real roots. It may have two distinct real roots, one real root (repeated), or no real roots, depending on the value of the discriminant ($D$).

3. Are there quadratic equations with no real roots?

Yes, there are quadratic equations with no real roots. This occurs when the discriminant ($D$) is less than zero, indicating that the equation has complex roots rather than real ones.

4. How is the discriminant used to determine the nature of roots?

The discriminant ($D$) is used to categorize the roots of a quadratic equation. If $D>0$, the equation has two distinct real roots. If $D=0$, the equation has one real root (repeated). If $D<0$, the equation has no real roots.

5. Can you provide an example of a quadratic equation with no real roots?

Certainly. Consider the quadratic equation �2+1=0x2+1=0. In this case, $a=1$, $b=0$, and $c=1$, leading to a discriminant $D=0-4(1)(1)=-4$, which is less than zero. Therefore, this equation has no real roots.

6. Why are quadratic equations important in mathematics?

Quadratic equations are important in mathematics because they appear in various real-world applications, including physics, engineering, economics, and computer science. They provide a fundamental framework for solving problems involving unknown quantities and are a cornerstone of algebraic understanding.

In Conclusion

In the world of quadratic equations, the existence of real roots depends on the value of the discriminant. Every quadratic equation has at least one real root when the discriminant ($D$) is greater than or equal to zero. However, there are cases where the discriminant is negative, leading to quadratic equations with no real roots. Understanding the discriminant is key to unraveling the mysteries of quadratic equations and their solutions.

So, the next time you encounter a quadratic equation, remember that whether it has real roots or not hinges on this mathematical parameter. It’s yet another fascinating aspect of the world of mathematics, where even seemingly simple equations can hold profound truths.