Select The Correct Answer.What Are The Roots Of This Quadratic Equation?$-10x^2 + 12x - 9 = 0$A. $x = -\frac{12}{5} \pm \frac{3i\sqrt{6}}{5}$B. $x = \frac{3}{10} \pm \frac{3i\sqrt{24}}{20}$C. $x = \frac{3}{5} \pm
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will delve into the world of quadratic equations and explore the roots of a given quadratic equation. We will use the quadratic formula to find the roots and compare the results with the given options.
What are Quadratic Equations?
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (usually x) 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. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.
The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
where a, b, and c are the coefficients of the quadratic equation. The quadratic formula is a general solution that works for all quadratic equations.
Solving the Given Quadratic Equation
Let's apply the quadratic formula to the given quadratic equation:
-10x^2 + 12x - 9 = 0
We can rewrite the equation in the standard form as:
10x^2 - 12x + 9 = 0
Now, we can identify the coefficients a, b, and c:
a = 10, b = -12, and c = 9
Substituting these values into the quadratic formula, we get:
x = (12 ± √((-12)^2 - 4(10)(9))) / (2(10))
Simplifying the expression, we get:
x = (12 ± √(144 - 360)) / 20
x = (12 ± √(-216)) / 20
x = (12 ± 6i√6) / 20
x = (3/5 ± 3i√6/5) / 2
x = (3/10 ± 3i√6/10)
Comparing the Results
Now, let's compare the results with the given options:
A. x = -12/5 ± 3i√6/5 B. x = 3/10 ± 3i√24/20 C. x = 3/5 ± 3i√6/5
Our solution matches option C, which is:
x = 3/5 ± 3i√6/5
Conclusion
In this article, we used the quadratic formula to solve a given quadratic equation. We identified the coefficients a, b, and c, and substituted them into the quadratic formula. We simplified the expression and obtained the roots of the equation. Our solution matched option C, which is:
x = 3/5 ± 3i√6/5
Introduction
Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we explored the roots of a given quadratic equation using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights into this fascinating topic.
Q: What is the difference between a quadratic equation and a linear equation?
A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable (usually x) is one. A quadratic equation, on the other hand, is a polynomial equation of degree two, which means the highest power of the variable (usually x) is two.
Q: How do I determine if a quadratic equation has real or complex roots?
A: To determine if a quadratic equation has real or complex roots, you need to examine the discriminant (b^2 - 4ac). If the discriminant is positive, the equation has two real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has two complex roots.
Q: What is the quadratic formula, and how do I use it to solve quadratic equations?
A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 - 4ac)) / 2a
To use the quadratic formula, you need to identify the coefficients a, b, and c in the quadratic equation. Then, substitute these values into the formula and simplify the expression to obtain the roots of the equation.
Q: Can I solve quadratic equations by factoring?
A: Yes, you can solve quadratic equations by factoring. However, factoring is not always possible, and it can be a challenging task. The quadratic formula is a more general solution that works for all quadratic equations.
Q: What is the difference between a quadratic equation and a polynomial equation of degree three or higher?
A: A polynomial equation of degree three or higher is called a cubic equation or a quartic equation, respectively. These equations have more complex roots and require more advanced techniques to solve.
Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?
A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you need to be careful when working with complex numbers and ensure that you follow the correct procedures.
Q: Are there any real-world applications of quadratic equations?
A: Yes, quadratic equations have many real-world applications in fields such as physics, engineering, economics, and computer science. For example, quadratic equations are used to model the motion of objects, optimize systems, and solve problems in machine learning.
Conclusion
In this article, we answered some frequently asked questions about quadratic equations and provided additional insights into this fascinating topic. We hope this article has provided a clear understanding of quadratic equations and their applications. Whether you are a student or a professional, quadratic equations are an essential tool for solving problems and making predictions in a wide range of fields.