Find The Discriminant Of The Quadratic Equation: ${ 10z^2 - 27z + 18 = 0 }$Discriminant = □

Mar 12, 2025 by ADMIN 93 views

**Finding the Discriminant of a Quadratic Equation**

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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 $az^2 + bz + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $z$ is the variable. The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation and is used to determine the nature of the roots of the equation. In this article, we will learn how to find the discriminant of a quadratic equation and discuss its significance.

What is the Discriminant?

The discriminant of a quadratic equation is denoted by the symbol $\Delta$ or $D$ and is calculated using the formula $\Delta = b^2 - 4ac$ . The discriminant is an important concept in algebra as it helps us determine the nature of the roots of the quadratic 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.

Calculating the Discriminant

To calculate the discriminant of a quadratic equation, we need to substitute the values of $a$ , $b$ , and $c$ into the formula $\Delta = b^2 - 4ac$ . Let's consider an example to illustrate this. Suppose we have the quadratic equation $10z^2 - 27z + 18 = 0$ . To find the discriminant, we need to substitute the values of $a$ , $b$ , and $c$ into the formula.

Step 1: Identify the Coefficients

The first step is to identify the coefficients of the quadratic equation. In this case, $a = 10$ , $b = -27$ , and $c = 18$ .

Step 2: Substitute the Coefficients into the Formula

Now that we have identified the coefficients, we can substitute them into the formula $\Delta = b^2 - 4ac$ . Plugging in the values, we get $\Delta = (-27)^2 - 4(10)(18)$ .

Step 3: Simplify the Expression

To simplify the expression, we need to evaluate the squares and the product. Evaluating the squares, we get $(-27)^2 = 729$ . Evaluating the product, we get $4(10)(18) = 720$ . Now, we can substitute these values back into the expression to get $\Delta = 729 - 720$ .

Step 4: Calculate the Discriminant

Finally, we can calculate the discriminant by subtracting 720 from 729. This gives us $\Delta = 9$ .

Conclusion

In this article, we learned how to find the discriminant of a quadratic equation. We discussed the significance of the discriminant and how it can be used to determine the nature of the roots of the equation. We also walked through an example of calculating the discriminant of a quadratic equation. By following these steps, you can easily calculate the discriminant of any quadratic equation.

Significance of the Discriminant

The discriminant is an important concept in algebra as it helps us determine the nature of the roots of the quadratic 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. This information is crucial in solving quadratic equations and is used in various applications in mathematics and science.

Real-World Applications

The discriminant has numerous real-world applications in mathematics and science. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to design stable structures. In economics, the discriminant is used to model the behavior of economic systems.

Conclusion

In conclusion, the discriminant is an important concept in algebra that helps us determine the nature of the roots of a quadratic equation. By calculating the discriminant, we can determine whether the equation has two distinct real roots, one real root, or no real roots. This information is crucial in solving quadratic equations and is used in various applications in mathematics and science.

Final Thoughts

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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 is used to determine 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 discriminant tell us about the roots of the equation?

A: The discriminant tells us whether the equation has two distinct real roots, one real root, or no real roots. 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.

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 and then use the following criteria:

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.

Q: Can I use the discriminant to solve quadratic equations?

A: Yes, you can use the discriminant to solve quadratic equations. If the discriminant is positive, you can use the quadratic formula to find the roots of the equation. If the discriminant is zero, you can use the fact that the equation has one real root to find the root. If the discriminant is negative, you can use the fact that the equation has no real roots to conclude that the equation has complex roots.

Q: What are some real-world applications of the discriminant?

A: The discriminant has numerous real-world applications in mathematics and science. For example, in physics, the discriminant is used to determine the stability of a system. In engineering, the discriminant is used to design stable structures. In economics, the discriminant is used to model the behavior of economic systems.

Q: Can I use the discriminant to determine the stability of a system?

A: Yes, you can use the discriminant to determine the stability of a system. If the discriminant is positive, the system is stable. If the discriminant is zero, the system is marginally stable. If the discriminant is negative, the system is unstable.

Q: Can I use the discriminant to design stable structures?

A: Yes, you can use the discriminant to design stable structures. If the discriminant is positive, the structure is stable. If the discriminant is zero, the structure is marginally stable. If the discriminant is negative, the structure is unstable.

Q: Can I use the discriminant to model the behavior of economic systems?

A: Yes, you can use the discriminant to model the behavior of economic systems. If the discriminant is positive, the system is stable. If the discriminant is zero, the system is marginally stable. If the discriminant is negative, the system is unstable.

Conclusion

Final Thoughts

In this article, we answered some frequently asked questions about the discriminant of a quadratic equation. We discussed the significance of the discriminant and how it can be used to determine the nature of the roots of the equation. We also provided some real-world applications of the discriminant. By understanding the discriminant, you can better solve quadratic equations and apply the concepts to real-world problems.