Solve The Equation: $\[ Y^2 + 12y - 44 = 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 the quadratic equation $y^2 + 12y - 44 = 0$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and factoring techniques.

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 example equation, $y^2 + 12y - 44 = 0$, we have $a = 1$, $b = 12$, and $c = -44$.

Step 1: Factor the Quadratic Equation

To solve the quadratic equation, we can try to factor it into the product of two binomials. This involves finding two numbers whose product is $ac$ (in this case, $1 \times -44 = -44$) and whose sum is $b$ (in this case, $12$). These numbers are $-11$ and $4$, since $-11 \times 4 = -44$ and $-11 + 4 = -7$. However, we need to find two numbers that add up to 12, and their product is -44.

Let's try to factor the equation by grouping the terms:

$$y^2 + 12y - 44 = (y^2 + 12y) - 44$$

Now, we can factor out a common factor of $y$ from the first two terms:

$$y^2 + 12y - 44 = y(y + 12) - 44$$

Next, we can try to factor the expression $y(y + 12) - 44$ by finding two numbers whose product is $-44$ and whose sum is $12$. These numbers are $-11$ and $4$, since $-11 \times 4 = -44$ and $-11 + 4 = -7$. However, we need to find two numbers that add up to 12, and their product is -44.

After trying different combinations, we find that the equation can be factored as:

$$y^2 + 12y - 44 = (y + 11)(y + 4)$$

Step 2: Solve for $y$

Now that we have factored the quadratic equation, we can set each factor equal to zero and solve for $y$:

$$(y + 11) = 0 \quad \text{or} \quad (y + 4) = 0$$

Solving for $y$ in each equation, we get:

$$y + 11 = 0 \quad \Rightarrow \quad y = -11$$

$$y + 4 = 0 \quad \Rightarrow \quad y = -4$$

Therefore, the solutions to the quadratic equation $y^2 + 12y - 44 = 0$ are $y = -11$ and $y = -4$.

Conclusion

Solving quadratic equations is an essential skill in mathematics, and factoring is a powerful technique for solving these equations. By breaking down the solution into manageable steps and using algebraic manipulations and factoring techniques, we can solve even the most challenging quadratic equations. In this article, we solved the quadratic equation $y^2 + 12y - 44 = 0$ using factoring, and we found the solutions to be $y = -11$ and $y = -4$.

Additional Resources

For more information on solving quadratic equations, including factoring and other techniques, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

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: There are several techniques for solving quadratic equations, including factoring, using the quadratic formula, and graphing.
• Q: What is the quadratic formula? A: The quadratic formula is a formula for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Glossary

• Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Factoring: A technique for solving quadratic equations by expressing the equation as the product of two binomials.
• Quadratic formula: A formula for solving quadratic equations of the form $ax^2 + bx + c = 0$.

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Quadratic Equations" by Khan Academy
  Quadratic Equations Q&A: Frequently Asked Questions =====================================================

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, including factoring, the quadratic formula, and graphing.

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. 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: There are several techniques for solving quadratic equations, including factoring, using the quadratic formula, and graphing. The technique you choose will depend on the specific equation and the information you are given.

Q: What is factoring?

A: Factoring is a technique for solving quadratic equations by expressing the equation as the product of two binomials. This involves finding two numbers whose product is $ac$ (in this case, $a \times c$) and whose sum is $b$ (in this case, $b$). These numbers are called the "factors" of the equation.

Q: How do I factor a quadratic equation?

A: To factor a quadratic equation, you need to find two numbers whose product is $ac$ and whose sum is $b$. These numbers are called the "factors" of the equation. You can use a variety of techniques to find the factors, including trial and error, using the quadratic formula, and graphing.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations of the form $ax^2 + bx + c = 0$. The formula is:

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

Q: How do I use the quadratic formula?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. You can then simplify the expression and solve for $x$.

Q: What is graphing?

A: Graphing is a technique for solving quadratic equations by graphing the equation on a coordinate plane. This involves plotting the points where the equation intersects the x-axis and the y-axis.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you need to plot the points where the equation intersects the x-axis and the y-axis. You can use a variety of techniques to find these points, including factoring, using the quadratic formula, and graphing.

Q: What are the advantages and disadvantages of each technique?

A: Each technique has its own advantages and disadvantages. Factoring is a simple and intuitive technique, but it can be difficult to use for complex equations. The quadratic formula is a powerful technique, but it can be difficult to use for equations with complex roots. Graphing is a visual technique, but it can be difficult to use for equations with complex graphs.

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 for extraneous solutions
• Not using the correct technique for the specific equation
• Not simplifying the expression before solving for $x$
• Not checking the solutions for complex roots

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, including factoring, the quadratic formula, and graphing. We hope that this article has been helpful in answering your questions and providing you with a better understanding of quadratic equations.

Additional Resources

For more information on solving quadratic equations, including factoring, the quadratic formula, and graphing, check out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Glossary

• Quadratic equation: A polynomial equation of degree two, which means the highest power of the variable is two.
• Factoring: A technique for solving quadratic equations by expressing the equation as the product of two binomials.
• Quadratic formula: A formula for solving quadratic equations of the form $ax^2 + bx + c = 0$.
• Graphing: A technique for solving quadratic equations by graphing the equation on a coordinate plane.

References

• "Algebra and Trigonometry" by Michael Sullivan
• "College Algebra" by James Stewart
• "Quadratic Equations" by Khan Academy