Solve The Equation. Y 2 = − 22 5 Y − 121 25 Y^2 = -\frac{22}{5} Y - \frac{121}{25} Y 2 = − 5 22 ​ Y − 25 121 ​ A. Y = 0 , Y = − 11 5 Y = 0, Y = -\frac{11}{5} Y = 0 , Y = − 5 11 ​ B. Y = 11 5 Y = \frac{11}{5} Y = 5 11 ​ C. Y = − 11 5 Y = -\frac{11}{5} Y = − 5 11 ​ D. Y = − 11 , Y = 11 Y = -11, Y = 11 Y = − 11 , Y = 11

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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, y2=−225y−12125y^2 = -\frac{22}{5} y - \frac{121}{25}, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, yy) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants, and xx is the variable. In our equation, yy is the variable, and the coefficients are a=1a = 1, b=225b = \frac{22}{5}, and c=12125c = \frac{121}{25}.

Solving the Equation

To solve the equation y2=−225y−12125y^2 = -\frac{22}{5} y - \frac{121}{25}, we can start by rearranging the terms to get:

y2+225y+12125=0y^2 + \frac{22}{5} y + \frac{121}{25} = 0

This is a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0, where a=1a = 1, b=225b = \frac{22}{5}, and c=12125c = \frac{121}{25}.

Factoring the Quadratic Equation

One method to solve quadratic equations is by factoring. However, in this case, the equation does not factor easily. Therefore, we will use the quadratic formula to find the solutions.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

In our equation, a=1a = 1, b=225b = \frac{22}{5}, and c=12125c = \frac{121}{25}. Plugging these values into the quadratic formula, we get:

y=−225±(225)2−4(1)(12125)2(1)y = \frac{-\frac{22}{5} \pm \sqrt{\left(\frac{22}{5}\right)^2 - 4(1)\left(\frac{121}{25}\right)}}{2(1)}

Simplifying the expression under the square root, we get:

y=−225±48425−484252y = \frac{-\frac{22}{5} \pm \sqrt{\frac{484}{25} - \frac{484}{25}}}{2}

The expression under the square root is zero, so we have:

y=−225±02y = \frac{-\frac{22}{5} \pm 0}{2}

This simplifies to:

y=−2210y = -\frac{22}{10}

Simplifying the Solution

We can simplify the solution further by dividing both the numerator and the denominator by their greatest common divisor, which is 2. This gives us:

y=−115y = -\frac{11}{5}

Conclusion

In this article, we solved the quadratic equation y2=−225y−12125y^2 = -\frac{22}{5} y - \frac{121}{25} using the quadratic formula. We found that the solution is y=−115y = -\frac{11}{5}. This is a key concept in mathematics, and understanding how to solve quadratic equations is essential for success in many fields.

Answer

The correct answer is:

  • C. y=−115y = -\frac{11}{5}

Additional Tips and Resources

  • To solve quadratic equations, you can use the quadratic formula or factoring.
  • The quadratic formula is a powerful tool for solving quadratic equations.
  • To simplify the solution, you can divide both the numerator and the denominator by their greatest common divisor.
  • For more information on quadratic equations, check out the following resources:
    • Khan Academy: Quadratic Equations
    • Mathway: Quadratic Equations
    • Wolfram Alpha: Quadratic Equations
      Quadratic Equations Q&A ==========================

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 (in this case, yy) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including:

  • Factoring: If the equation can be factored easily, you can solve it by finding the factors.
  • Quadratic Formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form ax2+bx+c=0ax^2 + bx + c = 0, the solutions are given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What is the quadratic formula?

A: The quadratic formula is a formula that gives the solutions to a quadratic equation. It is given by:

x=−b±b2−4ac2ax = \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 plug in the values of aa, bb, and cc into the formula. Then, simplify the expression under the square root and solve for xx.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula is a general method for solving quadratic equations, while factoring is a specific method that only works for certain types of quadratic equations. Factoring is often faster and easier than using the quadratic formula, but it only works if the equation can be factored easily.

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

A: Yes, the quadratic formula can be used to solve quadratic equations with complex solutions. In this case, the expression under the square root will be negative, and you will need to use the imaginary unit ii to simplify the solution.

Q: How do I simplify complex solutions?

A: To simplify complex solutions, you can use the following steps:

  • Simplify the expression under the square root.
  • Use the imaginary unit ii to simplify the solution.
  • Combine like terms and simplify the expression.

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

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

  • Not simplifying the expression under the square root.
  • Not using the correct values of aa, bb, and cc in the quadratic formula.
  • Not simplifying the solution after using the quadratic formula.

Q: How do I check my solutions?

A: To check your solutions, you can plug them back into the original equation and simplify. If the solutions are correct, the equation should be true.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

  • Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
  • Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
  • Economics: Quadratic equations are used to model economic systems and make predictions about future trends.

Q: How do I practice solving quadratic equations?

A: To practice solving quadratic equations, you can try the following:

  • Use online resources, such as Khan Academy or Mathway, to practice solving quadratic equations.
  • Work on problems from a textbook or worksheet.
  • Try solving quadratic equations on your own, without looking at the solutions.

Q: What are some common types of quadratic equations?

A: Some common types of quadratic equations include:

  • Quadratic equations with real solutions.
  • Quadratic equations with complex solutions.
  • Quadratic equations with rational solutions.
  • Quadratic equations with irrational solutions.

Q: How do I identify the type of quadratic equation?

A: To identify the type of quadratic equation, you can look at the expression under the square root. If it is positive, the equation has real solutions. If it is negative, the equation has complex solutions. If it is zero, the equation has rational solutions. If it is a perfect square, the equation has irrational solutions.