Nature of Roots Based on the Discriminant

Oct 13, 2025 by miodeya.com 62 views

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Introduction

In algebra, a quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation, and it plays a crucial role in determining the nature of the roots of the equation.

What is the Discriminant?

The discriminant of a quadratic equation is denoted by the symbol $\Delta$ or $D$ , and it is calculated using the formula $\Delta = b^2 - 4ac$ . The discriminant is a value that can be either positive, negative, or zero, and it determines the nature of the roots of the equation.

Nature of Roots Based on the Discriminant

If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root, which is also known as a repeated root. If the discriminant is negative, the equation has no real roots, and the roots are complex conjugates.

Calculating the Discriminant

To calculate the discriminant of a quadratic equation, we need to identify the values of $a$ , $b$ , and $c$ in the equation. Once we have these values, we can plug them into the formula $\Delta = b^2 - 4ac$ to calculate the discriminant.

Example: Calculating the Discriminant of a Quadratic Equation

Let's consider the quadratic equation $-2x^2 - 40x - 128 = 0$ . To calculate the discriminant, we need to identify the values of $a$ , $b$ , and $c$ in the equation.

$a = -2$
$b = -40$
$c = -128$

Now, we can plug these values into the formula $\Delta = b^2 - 4ac$ to calculate the discriminant.

$\Delta = (-40)^2 - 4(-2)(-128)$ $\Delta = 1600 - 1024$ $\Delta = 576$

Conclusion

In this article, we discussed the concept of the discriminant of a quadratic equation and how it can be calculated using the formula $\Delta = b^2 - 4ac$ . We also explored the nature of the roots of a quadratic equation based on the value of the discriminant. Finally, we calculated the discriminant of a specific quadratic equation using the formula.

Final Answer

The final answer is: $\boxed{576}$

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Frequently Asked Questions

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation, and it plays a crucial role in determining the nature of the roots of the equation.

Q: How is the discriminant calculated?

A: The discriminant is calculated using the formula $\Delta = b^2 - 4ac$ , where $a$ , $b$ , and $c$ are the coefficients of the quadratic equation.

Q: What does the value of the discriminant indicate?

A: The value of the discriminant indicates the nature of the roots of the equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root, which is also known as a repeated root. If the discriminant is negative, the equation has no real roots, and the roots are complex conjugates.

Q: How do I determine the nature of the roots of a quadratic equation?

A: To determine the nature of the roots of a quadratic equation, you need to calculate the discriminant using the formula $\Delta = b^2 - 4ac$ . Then, based on the value of the discriminant, you can determine the nature of the roots.

Q: What is the significance of the discriminant in quadratic equations?

A: The discriminant is significant in quadratic equations because it determines the nature of the roots of the equation. It helps us understand whether the equation has real roots, complex roots, or repeated roots.

Q: Can the discriminant be zero?

A: Yes, the discriminant can be zero. If the discriminant is zero, the equation has one real root, which is also known as a repeated root.

Q: Can the discriminant be negative?

A: Yes, the discriminant can be negative. If the discriminant is negative, the equation has no real roots, and the roots are complex conjugates.

Q: How do I calculate the discriminant of a quadratic equation with complex coefficients?

A: To calculate the discriminant of a quadratic equation with complex coefficients, you need to use the formula $\Delta = b^2 - 4ac$ , where $a$ , $b$ , and $c$ are the complex coefficients of the equation.

Q: Can the discriminant be a complex number?

A: Yes, the discriminant can be a complex number. If the discriminant is a complex number, the equation has complex roots.

Q: How do I determine the nature of the roots of a quadratic equation with complex coefficients?

A: To determine the nature of the roots of a quadratic equation with complex coefficients, you need to calculate the discriminant using the formula $\Delta = b^2 - 4ac$ . Then, based on the value of the discriminant, you can determine the nature of the roots.

Conclusion

In this article, we answered some frequently asked questions about the discriminant of a quadratic equation. We discussed the significance of the discriminant, how to calculate it, and how to determine the nature of the roots of a quadratic equation based on the value of the discriminant.

Final Answer

The final answer is: $\boxed{576}$