Solve The Equation: ${ 2y^2 - 6y - 3 = 0 }$

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. 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.

In this article, we will focus on solving the quadratic equation $2y^2 - 6y - 3 = 0$. This equation is a classic example of a quadratic equation, and it can be solved using various methods such as factoring, completing the square, and the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is a formula that can be used to find the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$. The quadratic formula is given by:

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

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic equation, and $x$ is the variable.

Applying the Quadratic Formula to the Equation $2y^2 - 6y - 3 = 0$

To solve the equation $2y^2 - 6y - 3 = 0$, we can use the quadratic formula. In this equation, $a = 2$, $b = -6$, and $c = -3$. Plugging these values into the quadratic formula, we get:

$$y = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(2)(-3)}}{2(2)}$$

Simplifying this expression, we get:

$$y = \frac{6 \pm \sqrt{36 + 24}}{4}$$

$$y = \frac{6 \pm \sqrt{60}}{4}$$

$$y = \frac{6 \pm 2\sqrt{15}}{4}$$

$$y = \frac{3 \pm \sqrt{15}}{2}$$

Simplifying the Solutions

The solutions of the equation $2y^2 - 6y - 3 = 0$ are given by:

$$y = \frac{3 + \sqrt{15}}{2}$$

$$y = \frac{3 - \sqrt{15}}{2}$$

These solutions can be simplified further by rationalizing the denominators. To rationalize the denominators, we multiply the numerator and denominator of each solution by the conjugate of the denominator.

Rationalizing the Denominators

To rationalize the denominators, we multiply the numerator and denominator of each solution by the conjugate of the denominator. The conjugate of the denominator is the same expression with the opposite sign.

For the first solution, we multiply the numerator and denominator by $\frac{3 - \sqrt{15}}{2}$:

$$y = \frac{3 + \sqrt{15}}{2} \times \frac{3 - \sqrt{15}}{2}$$

$$y = \frac{9 - 15}{4}$$

$$y = \frac{-6}{4}$$

$$y = -\frac{3}{2}$$

For the second solution, we multiply the numerator and denominator by $\frac{3 + \sqrt{15}}{2}$:

$$y = \frac{3 - \sqrt{15}}{2} \times \frac{3 + \sqrt{15}}{2}$$

$$y = \frac{9 - 15}{4}$$

$$y = \frac{-6}{4}$$

$$y = -\frac{3}{2}$$

Conclusion

In this article, we solved the quadratic equation $2y^2 - 6y - 3 = 0$ using the quadratic formula. We applied the quadratic formula to the equation and simplified the solutions to get the final answers. We also rationalized the denominators of the solutions to get the final answers in the simplest form.

Final Answers

The final answers to the equation $2y^2 - 6y - 3 = 0$ are:

$$y = -\frac{3}{2}$$

$$y = -\frac{3}{2}$$

These are the solutions to the equation, and they can be used to solve various problems in mathematics and other fields.

Applications of Quadratic Equations

Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the applications of quadratic equations include:

• Projectile Motion: Quadratic equations are used to model the motion of projectiles under the influence of gravity.
• Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum values.
• Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
• Economics: Quadratic equations are used in economics to model the behavior of economic systems and make predictions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$. In this article, we solved the quadratic equation $2y^2 - 6y - 3 = 0$ using the quadratic formula and simplified the solutions to get the final answers. We also rationalized the denominators of the solutions to get the final answers in the simplest form.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In our previous article, we solved the quadratic equation $2y^2 - 6y - 3 = 0$ using the quadratic formula and simplified the solutions to get the final answers. In this article, we will answer some frequently asked questions about quadratic equations and provide additional information to help you understand this concept better.

Q1: What is a quadratic equation?

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.

Q2: How do I solve a quadratic equation?

There are several methods to solve a quadratic equation, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$.

Q3: What is the quadratic formula?

The quadratic formula is a formula that can be used to find the solutions of a quadratic equation in the form of $ax^2 + bx + c = 0$. The quadratic formula is given by:

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

Q4: How do I apply the quadratic formula to a quadratic equation?

To apply the quadratic formula to a quadratic equation, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, you can plug these values into the quadratic formula and simplify the expression to get the solutions.

Q5: What are the solutions of a quadratic equation?

The solutions of a quadratic equation are the values of the variable that satisfy the equation. In other words, they are the values of the variable that make the equation true.

Q6: How do I simplify the solutions of a quadratic equation?

To simplify the solutions of a quadratic equation, you can use various techniques such as rationalizing the denominators, combining like terms, and factoring.

Q7: What are the applications of quadratic equations?

Quadratic equations have numerous applications in various fields such as physics, engineering, and economics. Some of the applications of quadratic equations include:

• Projectile Motion: Quadratic equations are used to model the motion of projectiles under the influence of gravity.
• Optimization: Quadratic equations are used to optimize functions and find the maximum or minimum values.
• Signal Processing: Quadratic equations are used in signal processing to filter signals and remove noise.
• Economics: Quadratic equations are used in economics to model the behavior of economic systems and make predictions.

Q8: How do I use quadratic equations in real-life situations?

Quadratic equations can be used in various real-life situations such as:

• Designing a Trajectory: Quadratic equations can be used to design a trajectory for a projectile, such as a rocket or a ball.
• Optimizing a Function: Quadratic equations can be used to optimize a function, such as the cost of a product or the time it takes to complete a task.
• Filtering Signals: Quadratic equations can be used to filter signals and remove noise in signal processing.
• Modeling Economic Systems: Quadratic equations can be used to model the behavior of economic systems and make predictions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields. In this article, we answered some frequently asked questions about quadratic equations and provided additional information to help you understand this concept better. We hope that this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Final Tips

Here are some final tips to help you understand quadratic equations better:

• Practice, Practice, Practice: The more you practice solving quadratic equations, the more comfortable you will become with the concept.
• Use Online Resources: There are many online resources available that can help you learn quadratic equations, such as video tutorials, practice problems, and interactive simulations.
• Seek Help When Needed: Don't be afraid to seek help when you need it. Ask your teacher, a tutor, or a classmate for help if you are struggling with quadratic equations.
• Stay Motivated: Learning quadratic equations can be challenging, but it is also rewarding. Stay motivated by setting goals for yourself and celebrating your progress.

We hope that this article has been helpful in providing you with a better understanding of quadratic equations. Good luck with your studies!