Solve For $y$:$y^2 + 11y + 24 = 0$

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 quadratic equation of the form $y^2 + 11y + 24 = 0$. We will use the quadratic formula and factorization methods to find the value of y.

What is a Quadratic Equation?

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 case, the equation is $y^2 + 11y + 24 = 0$, where a = 1, b = 11, and c = 24.

The Quadratic Formula

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

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

where x is the variable, and a, b, and c are the constants from the quadratic equation. In our case, we can plug in the values of a, b, and c to get:

$$y = \frac{-11 \pm \sqrt{11^2 - 4(1)(24)}}{2(1)}$$

Simplifying the Quadratic Formula

To simplify the quadratic formula, we need to calculate the value of the expression inside the square root. This is given by:

$$b^2 - 4ac = 11^2 - 4(1)(24) = 121 - 96 = 25$$

Now, we can plug this value back into the quadratic formula:

$$y = \frac{-11 \pm \sqrt{25}}{2(1)}$$

Solving for y

To solve for y, we need to simplify the expression inside the square root. Since the square root of 25 is 5, we can write:

$$y = \frac{-11 \pm 5}{2(1)}$$

Now, we have two possible solutions for y:

$$y_1 = \frac{-11 + 5}{2(1)} = \frac{-6}{2} = -3$$

$$y_2 = \frac{-11 - 5}{2(1)} = \frac{-16}{2} = -8$$

Checking the Solutions

To check our solutions, we can plug them back into the original equation:

$$y^2 + 11y + 24 = 0$$

For $y_1 = -3$, we get:

$$(-3)^2 + 11(-3) + 24 = 9 - 33 + 24 = 0$$

This confirms that $y_1 = -3$ is a valid solution.

For $y_2 = -8$, we get:

$$(-8)^2 + 11(-8) + 24 = 64 - 88 + 24 = 0$$

This confirms that $y_2 = -8$ is also a valid solution.

Conclusion

In this article, we solved a quadratic equation of the form $y^2 + 11y + 24 = 0$ using the quadratic formula and factorization methods. We found two possible solutions for y, which were $y_1 = -3$ and $y_2 = -8$. We then checked our solutions by plugging them back into the original equation, and confirmed that both solutions are valid.

Tips and Tricks

• When solving quadratic equations, it's essential to check your solutions by plugging them back into the original equation.
• The quadratic formula can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$, where a, b, and c are constants.
• Factorization methods can also be used to solve quadratic equations, but they may not always be possible.

Common Quadratic Equations

• $x^2 + 4x + 4 = 0$
• $x^2 - 6x + 8 = 0$
• $x^2 + 2x - 6 = 0$

Quadratic Equation Formula

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

Quadratic Equation Examples

• $x^2 + 5x + 6 = 0$
• $x^2 - 3x - 4 = 0$
• $x^2 + 2x - 3 = 0$

Quadratic Equation Solutions

• $x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
• $x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$
  Quadratic Equation Q&A: Frequently Asked Questions and Answers ================================================================

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In this article, we will answer some of the most frequently asked questions about quadratic equations, including how to solve them, what the quadratic formula is, and how to check solutions.

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: What is the quadratic formula?

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

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

where x is the variable, and a, b, and c are the constants from the quadratic equation.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to plug in the values of a, b, and c from the quadratic equation into the formula. Then, simplify the expression inside the square root, and solve for x.

Q: What is the expression inside the square root in the quadratic formula?

A: The expression inside the square root in the quadratic formula is given by:

$$b^2 - 4ac$$

This expression is also known as the discriminant.

Q: What is the discriminant?

A: The discriminant is the expression inside the square root in the quadratic formula. It is given by:

$$b^2 - 4ac$$

The discriminant can be positive, negative, or zero.

Q: What happens if the discriminant is positive?

A: If the discriminant is positive, then the quadratic equation has two distinct real solutions.

Q: What happens if the discriminant is negative?

A: If the discriminant is negative, then the quadratic equation has no real solutions.

Q: What happens if the discriminant is zero?

A: If the discriminant is zero, then the quadratic equation has one real solution.

Q: How do I check my solutions to a quadratic equation?

A: To check your solutions to a quadratic equation, you need to plug them back into the original equation. If the solutions satisfy the equation, then they are valid.

Q: What are some common quadratic equations?

A: Some common quadratic equations include:

• $x^2 + 4x + 4 = 0$
• $x^2 - 6x + 8 = 0$
• $x^2 + 2x - 6 = 0$

Q: What are some tips for solving quadratic equations?

A: Some tips for solving quadratic equations include:

• Use the quadratic formula to solve quadratic equations.
• Check your solutions by plugging them back into the original equation.
• Use factorization methods to solve quadratic equations.
• Simplify the expression inside the square root in the quadratic formula.

Conclusion

In this article, we answered some of the most frequently asked questions about quadratic equations, including how to solve them, what the quadratic formula is, and how to check solutions. We also provided some tips for solving quadratic equations and listed some common quadratic equations.

Frequently Asked Questions

• Q: What is a quadratic equation?
• A: A quadratic equation is a polynomial equation of degree two.
• Q: What is the quadratic formula?
• A: The quadratic formula is a powerful tool for solving quadratic equations.
• Q: How do I use the quadratic formula to solve a quadratic equation?
• A: To use the quadratic formula, you need to plug in the values of a, b, and c from the quadratic equation into the formula.
• Q: What is the expression inside the square root in the quadratic formula?
• A: The expression inside the square root in the quadratic formula is given by $b^2 - 4ac$.

Quadratic Equation Formula

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

Quadratic Equation Examples

• $x^2 + 5x + 6 = 0$
• $x^2 - 3x - 4 = 0$
• $x^2 + 2x - 3 = 0$

Quadratic Equation Solutions

• $x_1 = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
• $x_2 = \frac{-b - \sqrt{b^2 - 4ac}}{2a}$