Percy Solved The Equation X 2 + 7 X + 12 = 12 X^2 + 7x + 12 = 12 X 2 + 7 X + 12 = 12 . His Work Is Shown Below. Is Percy Correct? Explain.1. $(x+3)(x+4) = 12$2. X + 3 = 12 X+3 = 12 X + 3 = 12 Or $x+4 = 12$3. X = 9 X = 9 X = 9 Or X = 8 X = 8 X = 8 Is Percy Correct? Explain.

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will explore the steps involved in solving quadratic equations and examine the work of Percy, a student who attempted to solve the equation $x^2 + 7x + 12 = 12$. We will analyze his work and determine whether he is correct or not.

Understanding Quadratic Equations

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. In the given equation, $x^2 + 7x + 12 = 12$, we can rewrite it as $x^2 + 7x + 12 - 12 = 0$, which simplifies to $x^2 + 7x = 0$.

Step 1: Factoring the Quadratic Equation

Percy's first step is to factor the quadratic equation $(x+3)(x+4) = 12$. However, this is not a correct step. The correct step would be to factor the quadratic equation $x^2 + 7x = 0$. We can factor out the greatest common factor (GCF), which is $x$. This gives us $x(x + 7) = 0$.

Step 2: Setting Up the Equations

Percy's next step is to set up the equations $x+3 = 12$ or $x+4 = 12$. However, this is not a correct step. The correct step would be to set up the equations $x(x + 7) = 0$. We can rewrite this as $x = 0$ or $x + 7 = 0$.

Step 3: Solving the Equations

Percy's final step is to solve the equations $x = 9$ or $x = 8$. However, this is not a correct step. The correct solutions are $x = 0$ or $x = -7$.

Conclusion

In conclusion, Percy's work is incorrect. He failed to factor the quadratic equation correctly and set up the equations incorrectly. The correct solutions to the equation $x^2 + 7x + 12 = 12$ are $x = 0$ or $x = -7$.

Tips for Solving Quadratic Equations

1. Factor the quadratic equation: Factor out the greatest common factor (GCF) and then factor the remaining expression.
2. Set up the equations: Set up the equations by setting each factor equal to zero.
3. Solve the equations: Solve the equations by isolating the variable.
4. Check the solutions: Check the solutions by plugging them back into the original equation.

Common Mistakes to Avoid

1. Not factoring the quadratic equation: Failing to factor the quadratic equation can lead to incorrect solutions.
2. Setting up the equations incorrectly: Setting up the equations incorrectly can lead to incorrect solutions.
3. Not checking the solutions: Failing to check the solutions can lead to incorrect answers.

Real-World Applications

Quadratic equations have many real-world applications, including:

1. Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
2. Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
3. Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Conclusion

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

1. Factor the quadratic equation: Factor out the greatest common factor (GCF) and then factor the remaining expression.
2. Set up the equations: Set up the equations by setting each factor equal to zero.
3. Solve the equations: Solve the equations by isolating the variable.
4. Check the solutions: Check the solutions by plugging them back into the original equation.

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

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

1. Not factoring the quadratic equation: Failing to factor the quadratic equation can lead to incorrect solutions.
2. Setting up the equations incorrectly: Setting up the equations incorrectly can lead to incorrect solutions.
3. Not checking the solutions: Failing to check the solutions can lead to incorrect answers.

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

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

1. Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
2. Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
3. Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.

Q: How do I determine the number of solutions to a quadratic equation?

A: To determine the number of solutions to a quadratic equation, you can use the following methods:

1. Graphing: Graph the quadratic equation to see if it intersects the x-axis at one or two points.
2. Factoring: Factor the quadratic equation to see if it can be factored into two binomials.
3. Using the discriminant: Use the discriminant to determine if the quadratic equation has one or two solutions.

Q: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is used to determine if the quadratic equation has one or two solutions.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you can use the following formula:

$$\Delta = b^2 - 4ac$$

Where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What does the discriminant tell me?

A: The discriminant tells you if the quadratic equation has one or two solutions. If the discriminant is positive, the quadratic equation has two solutions. If the discriminant is zero, the quadratic equation has one solution. If the discriminant is negative, the quadratic equation has no real solutions.

Q: How do I use the discriminant to determine the number of solutions?

A: To use the discriminant to determine the number of solutions, you can follow these steps:

1. Calculate the discriminant: Calculate the discriminant using the formula $\Delta = b^2 - 4ac$.
2. Determine the number of solutions: If the discriminant is positive, the quadratic equation has two solutions. If the discriminant is zero, the quadratic equation has one solution. If the discriminant is negative, the quadratic equation has no real solutions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. By following the steps outlined in this article, students can solve quadratic equations correctly and avoid common mistakes. The real-world applications of quadratic equations make them an essential part of mathematics and science.