What Is The Discriminant Of The Quadratic Equation $x^2 + 2x + 2 = 0$?A. 4 B. 12 C. -4 D. -12

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It has the general form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $a$ cannot be equal to zero. The quadratic equation is a fundamental concept in algebra and is used to model various real-world situations, such as the trajectory of a projectile, the motion of an object under the influence of gravity, and the growth of a population.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

The quadratic formula involves the discriminant, which 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 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.

The Quadratic Equation $x^2 + 2x + 2 = 0$

We are given the quadratic equation $x^2 + 2x + 2 = 0$. To find the discriminant, we need to identify the values of $a$, $b$, and $c$. In this case, $a = 1$, $b = 2$, and $c = 2$.

Calculating the Discriminant

The discriminant is given by the expression $b^2 - 4ac$. Plugging in the values of $a$, $b$, and $c$, we get:

$b^2 - 4ac = 2^2 - 4(1)(2)$ $b^2 - 4ac = 4 - 8$ $b^2 - 4ac = -4$

Conclusion

The discriminant of the quadratic equation $x^2 + 2x + 2 = 0$ is $-4$. This means that the equation has no real roots.

Understanding the Significance of the Discriminant

The discriminant is a crucial concept in algebra, and its significance extends beyond solving quadratic equations. The discriminant can be used to determine the nature of the roots of a quadratic equation, which is essential in various fields, such as physics, engineering, and economics.

Real-World Applications of the Discriminant

The discriminant has numerous real-world applications, including:

• Physics: The discriminant is used to determine the trajectory of a projectile under the influence of gravity.
• Engineering: The discriminant is used to design and optimize systems, such as bridges and buildings.
• Economics: The discriminant is used to model and analyze economic systems, such as supply and demand.

Conclusion

In conclusion, the discriminant is a fundamental concept in algebra that plays a crucial role in solving quadratic equations. The discriminant determines the nature of the roots of a quadratic equation, which is essential in various fields. The discriminant has numerous real-world applications, including physics, engineering, and economics.

Final Answer

The final answer is $\boxed{-4}$.

Discussion

The discriminant is a powerful tool for solving quadratic equations. However, it can be challenging to calculate the discriminant, especially for complex quadratic equations. In such cases, it is essential to use the quadratic formula and the discriminant to determine the nature of the roots.

Tips and Tricks

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It involves the discriminant, which determines the nature of the roots.
• Calculate the discriminant: The discriminant is given by the expression $b^2 - 4ac$. Plugging in the values of $a$, $b$, and $c$, we can calculate the discriminant.
• Determine the nature of the roots: The discriminant determines the nature of the roots of a 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.

Conclusion

In conclusion, the discriminant is a fundamental concept in algebra that plays a crucial role in solving quadratic equations. The discriminant determines the nature of the roots of a quadratic equation, which is essential in various fields. The discriminant has numerous real-world applications, including physics, engineering, and economics.

Final Answer

The final answer is $\boxed{-4}$.

References

• Algebra: A comprehensive textbook on algebra, including quadratic equations and the discriminant.
• Physics: A textbook on physics, including the trajectory of a projectile under the influence of gravity.
• Engineering: A textbook on engineering, including the design and optimization of systems.
• Economics: A textbook on economics, including the modeling and analysis of economic systems.

Conclusion

In conclusion, the discriminant is a fundamental concept in algebra that plays a crucial role in solving quadratic equations. The discriminant determines the nature of the roots of a quadratic equation, which is essential in various fields. The discriminant has numerous real-world applications, including physics, engineering, and economics.

Final Answer

The final answer is $\boxed{-4}$.

Introduction

The quadratic equation discriminant is a fundamental concept in algebra that plays a crucial role in solving quadratic equations. In this article, we will answer some of the most frequently asked questions about the quadratic equation discriminant.

Q1: What is the discriminant of a quadratic equation?

A1: The discriminant of a quadratic equation is the expression under the square root in the quadratic formula, given by $b^2 - 4ac$. It determines the nature of the roots of the quadratic equation.

Q2: How do I calculate the discriminant of a quadratic equation?

A2: To calculate the discriminant, you need to identify the values of $a$, $b$, and $c$ in the quadratic equation. Then, plug these values into the expression $b^2 - 4ac$ to get the discriminant.

Q3: What does the discriminant tell me about the roots of a quadratic equation?

A3: The discriminant determines the nature of the roots of a 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.

Q4: How do I use the discriminant to solve a quadratic equation?

A4: To use the discriminant to solve a quadratic equation, you need to first calculate the discriminant. Then, use the quadratic formula to find the roots of the equation. The quadratic formula is given by:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q5: What are some real-world applications of the discriminant?

A5: The discriminant has numerous real-world applications, including:

• Physics: The discriminant is used to determine the trajectory of a projectile under the influence of gravity.
• Engineering: The discriminant is used to design and optimize systems, such as bridges and buildings.
• Economics: The discriminant is used to model and analyze economic systems, such as supply and demand.

Q6: Can I use the discriminant to solve quadratic equations with complex roots?

A6: Yes, you can use the discriminant to solve quadratic equations with complex roots. If the discriminant is negative, the equation has complex roots. In this case, you can use the quadratic formula to find the complex roots.

Q7: How do I determine the nature of the roots of a quadratic equation without using the discriminant?

A7: You can determine the nature of the roots of a quadratic equation without using the discriminant by using the quadratic formula and checking the sign of the expression under the square root.

Q8: Can I use the discriminant to solve quadratic equations with rational roots?

A8: Yes, you can use the discriminant to solve quadratic equations with rational roots. If the discriminant is a perfect square, the equation has rational roots.

Q9: How do I use the discriminant to find the sum and product of the roots of a quadratic equation?

A9: You can use the discriminant to find the sum and product of the roots of a quadratic equation by using the following formulas:

• Sum of the roots: $-\frac{b}{a}$
• Product of the roots: $\frac{c}{a}$

Q10: Can I use the discriminant to solve quadratic equations with repeated roots?

A10: Yes, you can use the discriminant to solve quadratic equations with repeated roots. If the discriminant is zero, the equation has repeated roots.

Conclusion

In conclusion, the discriminant is a fundamental concept in algebra that plays a crucial role in solving quadratic equations. We hope that this Q&A article has helped you to understand the discriminant and how to use it to solve quadratic equations.

Final Answer

The final answer is $\boxed{-4}$.

References

• Algebra: A comprehensive textbook on algebra, including quadratic equations and the discriminant.
• Physics: A textbook on physics, including the trajectory of a projectile under the influence of gravity.
• Engineering: A textbook on engineering, including the design and optimization of systems.
• Economics: A textbook on economics, including the modeling and analysis of economic systems.

Conclusion

In conclusion, the discriminant is a fundamental concept in algebra that plays a crucial role in solving quadratic equations. We hope that this Q&A article has helped you to understand the discriminant and how to use it to solve quadratic equations.