Calculating the Discriminant

Oct 13, 2025 by windailys.com 2 views

The quadratic equation is a fundamental concept in mathematics, and it is used to model various real-world situations. In this article, we will focus on the quadratic equation $0 = 2x^2 + x - 3$ and explore the concept of the discriminant. The discriminant is a value that can be calculated from the coefficients of the quadratic equation, and it plays a crucial role in determining the nature of the roots of the equation.

What is the Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means that 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. In the given equation $0 = 2x^2 + x - 3$ , the coefficients are $a = 2$ , $b = 1$ , and $c = -3$ .

The Discriminant Formula

The discriminant is calculated using the formula $b^2 - 4ac$ . This formula is derived from the quadratic formula, which is used to find the roots of a quadratic equation. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . The discriminant is the value under the square root in the quadratic formula.

Calculating the Discriminant

To calculate the discriminant, we need to substitute the values of $a$ , $b$ , and $c$ into the formula $b^2 - 4ac$ . In this case, $a = 2$ , $b = 1$ , and $c = -3$ . Plugging these values into the formula, we get:

$b^2 - 4ac = (1)^2 - 4(2)(-3)$ $b^2 - 4ac = 1 + 24$ $b^2 - 4ac = 25$

Interpreting the Result

The calculated value of the discriminant is $25$ . This means that the discriminant is positive, which indicates that the quadratic equation has two distinct real roots.

Conclusion

In conclusion, the value of the discriminant for the quadratic equation $0 = 2x^2 + x - 3$ is $25$ . This value is calculated using the formula $b^2 - 4ac$ , and it indicates that the quadratic equation has two distinct real roots.

Answer

The correct answer is C. 25.

Additional Information

The discriminant is a crucial concept in mathematics, and it has various applications in different fields. It is used to determine the nature of the roots of a quadratic equation, and it is also used in the quadratic formula to find the roots of the equation.

Real-World Applications

The quadratic equation and the discriminant have various real-world applications. For example, they are used in physics to model the motion of objects, in engineering to design buildings and bridges, and in economics to model the behavior of markets.

Final Thoughts

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about the quadratic equation and the discriminant.

Q: What is the quadratic equation?

A: The quadratic equation is a polynomial equation of degree two, which means that 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.

Q: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of the quadratic equation, and it plays a crucial role in determining the nature of the roots of the equation. The discriminant is calculated using the formula $b^2 - 4ac$ .

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to substitute the values of $a$ , $b$ , and $c$ into the formula $b^2 - 4ac$ . For example, if the quadratic equation is $0 = 2x^2 + x - 3$ , the discriminant would be calculated as follows:

$b^2 - 4ac = (1)^2 - 4(2)(-3)$ $b^2 - 4ac = 1 + 24$ $b^2 - 4ac = 25$

Q: What does the discriminant tell me about the roots of the equation?

A: The discriminant tells you about 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. If the discriminant is negative, the equation has no real roots.