Solve For { Y $} . . . { \frac{1}{2} Y^2 = 12 - Y \}

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 focus on solving a specific quadratic equation, $\frac{1}{2} y^2 = 12 - y$, to find the value of $y$. We will break down the solution into manageable steps, using algebraic techniques and mathematical concepts to arrive at the final answer.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $y$) 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 our equation, $\frac{1}{2} y^2 = 12 - y$, we can rewrite it in the standard form as $\frac{1}{2} y^2 + y - 12 = 0$.

Rearranging the Equation

To solve the equation, we need to isolate the variable $y$. We can start by rearranging the equation to get all the terms on one side. This will give us $\frac{1}{2} y^2 + y - 12 = 0$. We can then combine like terms to simplify the equation.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In our equation, $a = \frac{1}{2}$, $b = 1$, and $c = -12$. Plugging these values into the quadratic formula, we get:

$$y = \frac{-1 \pm \sqrt{1^2 - 4(\frac{1}{2})(-12)}}{2(\frac{1}{2})}$$

Simplifying the Expression

We can simplify the expression under the square root by evaluating the expression inside the parentheses. This gives us:

$$y = \frac{-1 \pm \sqrt{1 + 24}}{1}$$

$$y = \frac{-1 \pm \sqrt{25}}{1}$$

Evaluating the Square Root

The square root of 25 is 5, so we can simplify the expression further:

$$y = \frac{-1 \pm 5}{1}$$

Finding the Two Possible Solutions

We now have two possible solutions for $y$, which are given by:

$$y = \frac{-1 + 5}{1} = 4$$

$$y = \frac{-1 - 5}{1} = -6$$

Conclusion

In this article, we solved the quadratic equation $\frac{1}{2} y^2 = 12 - y$ to find the value of $y$. We used algebraic techniques and mathematical concepts to arrive at the final answer, which is $y = 4$ or $y = -6$. This demonstrates the importance of quadratic equations in mathematics and their applications in various fields.

Additional Tips and Tricks

• When solving quadratic equations, it's essential to check the solutions by plugging them back into the original equation.
• The quadratic formula can be used to solve quadratic equations with complex coefficients.
• Quadratic equations can be used to model real-world problems, such as the motion of objects under the influence of gravity.

Final Thoughts

Solving quadratic equations is a fundamental skill that requires practice and patience. By following the steps outlined in this article, you can develop the skills and confidence to tackle more complex mathematical problems. Remember to always check your solutions and to use the quadratic formula as a powerful tool for solving quadratic equations.

Frequently Asked Questions

• Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.
• Q: How do I solve a quadratic equation? A: You can use the quadratic formula, which states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• Q: What are the two possible solutions for $y$ in the equation $\frac{1}{2} y^2 = 12 - y$? A: The two possible solutions for $y$ are $y = 4$ and $y = -6$.

References

• [1] "Quadratic Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Quadratic Formula" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-formula/x2f-quadratic-formula-article
• [3] "Solving Quadratic Equations" by Purplemath. Retrieved from https://www.purplemath.com/modules/solvquad.htm
  

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 answer some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $y$) 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: How do I solve a quadratic equation?

A: You can use the quadratic formula, which states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. Alternatively, you can factor the equation, complete the square, or use other algebraic techniques to solve the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form $ax^2 + bx + c = 0$, the solutions are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the equation. Then, plug these values into the formula, and simplify the expression to find the solutions.

Q: What are the two possible solutions for $y$ in the equation $\frac{1}{2} y^2 = 12 - y$?

A: The two possible solutions for $y$ are $y = 4$ and $y = -6$.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation. If the equation is true for both solutions, then you have found the correct solutions.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, you need to be careful when working with complex numbers.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not checking the solutions
• Not simplifying the expression under the square root
• Not using the correct values of $a$, $b$, and $c$
• Not following the correct order of operations

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the x-intercepts and the vertex of the parabola to plot the graph. Alternatively, you can use a graphing calculator or software to graph the equation.

Q: What is the vertex of a quadratic equation?

A: The vertex of a quadratic equation is the point on the parabola where the graph changes direction. It is given by the formula $x = \frac{-b}{2a}$.

Q: How do I find the x-intercepts of a quadratic equation?

A: To find the x-intercepts of a quadratic equation, set the equation equal to zero and solve for $x$. The x-intercepts are the points where the graph crosses the x-axis.

Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant is the expression under the square root in the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can I use quadratic equations to model real-world problems?

A: Yes, quadratic equations can be used to model real-world problems, such as the motion of objects under the influence of gravity, the trajectory of a projectile, or the growth of a population.

Q: What are some common applications of quadratic equations?

A: Some common applications of quadratic equations include:

• Physics: to model the motion of objects under the influence of gravity
• Engineering: to design and optimize systems
• Economics: to model the growth of a population or the behavior of a market
• Computer Science: to solve problems in computer graphics and game development

Q: How do I use quadratic equations in real-world problems?

A: To use quadratic equations in real-world problems, you need to identify the variables and the relationships between them. Then, you can use the quadratic formula or other algebraic techniques to solve the equation and find the solutions.

Q: What are some tips for solving quadratic equations?

A: Some tips for solving quadratic equations include:

• Check your solutions
• Simplify the expression under the square root
• Use the correct values of $a$, $b$, and $c$
• Follow the correct order of operations
• Use a graphing calculator or software to graph the equation

Q: How do I practice solving quadratic equations?

A: To practice solving quadratic equations, you can try solving problems on your own or using online resources, such as worksheets or practice exams. You can also work with a tutor or a study group to practice solving quadratic equations.

Q: What are some resources for learning more about quadratic equations?

A: Some resources for learning more about quadratic equations include:

• Online tutorials and videos
• Textbooks and workbooks
• Online courses and degree programs
• Study groups and tutoring services

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have answered some of the most frequently asked questions about quadratic equations, providing a comprehensive guide to help you understand and solve these equations. Whether you are a student or a professional, we hope that this article has been helpful in your journey to learn more about quadratic equations.