Not rearranging the equation

Oct 13, 2025 by miodeya.com 132 views

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, $p^2 - 12p = -20$ , and provide a step-by-step guide on how to do it.

What are Quadratic Equations?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $p$ ) 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.

The Given Equation

The given equation is $p^2 - 12p = -20$ . To solve this equation, we need to isolate the variable $p$ .

Step 1: Rearrange the Equation

The first step is to rearrange the equation to get all the terms on one side of the equation. We can do this by adding $20$ to both sides of the equation.

p^2 - 12p + 20 = 0

Step 2: Factor the Equation

Now that we have the equation in the form $p^2 - 12p + 20 = 0$ , we can try to factor it. Factoring a quadratic equation involves finding two numbers whose product is the constant term ( $20$ ) and whose sum is the coefficient of the linear term ( $-12$ ).

(p - 10)(p - 2) = 0

Step 3: Solve for $p$

Now that we have factored the equation, we can set each factor equal to zero and solve for $p$ .

p - 10 = 0 \Rightarrow p = 10
p - 2 = 0 \Rightarrow p = 2

Conclusion

In this article, we have solved the quadratic equation $p^2 - 12p = -20$ using the method of factoring. We first rearranged the equation to get all the terms on one side, then factored the equation, and finally solved for $p$ by setting each factor equal to zero.

Why is Solving Quadratic Equations Important?

Solving quadratic equations is an essential skill in mathematics, and it has numerous applications in various fields, such as physics, engineering, and economics. Some of the reasons why solving quadratic equations is important include:

Problem-solving: Quadratic equations are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.
Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
Data analysis: Quadratic equations are used to analyze data, such as fitting curves to data points.

Common Mistakes to Avoid

When solving quadratic equations, there are several common mistakes to avoid, including:

Not rearranging the equation: Failing to rearrange the equation to get all the terms on one side can lead to incorrect solutions.
Not factoring the equation: Failing to factor the equation can make it difficult to solve for the variable.
Not checking the solutions: Failing to check the solutions can lead to incorrect answers.

Tips and Tricks

Here are some tips and tricks to help you solve quadratic equations:

Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Check the solutions: Always check the solutions to make sure they are correct.
Use technology: Technology, such as calculators and computer software, can be used to solve quadratic equations.

Conclusion

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about quadratic 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, $p$ ) 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: To solve a quadratic equation, you can use the method of factoring, the quadratic formula, or graphing. The method of factoring involves finding two numbers whose product is the constant term ( $c$ ) and whose sum is the coefficient of the linear term ( $b$ ). The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . Graphing involves plotting the equation on a graph and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ . This formula can be used to find the solutions to a quadratic equation when the equation cannot be factored.

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. Then, simplify the expression and solve for $x$ .

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared variable, while a linear equation does not.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the graph of a quadratic equation is a parabola, which has at most two x-intercepts.

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 discriminant, which is given by $b^2 - 4ac$ . 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: What is the discriminant?

A: The discriminant is a value that can be calculated from the coefficients of a quadratic equation. It is given by $b^2 - 4ac$ . The discriminant can be used to determine the number of solutions to a quadratic equation.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to plug in the values of $a$ , $b$ , and $c$ into the formula $b^2 - 4ac$ . Then, simplify the expression.

Q: What is the significance of the discriminant?

A: The discriminant is significant because it can be used to determine the number of solutions to a quadratic 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.

Conclusion

In conclusion, quadratic equations are an essential part of mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the concepts and techniques outlined in this article, you can solve quadratic equations with ease. Remember to always check the solutions and use technology to help you solve the equation. With practice and patience, you can become proficient in solving quadratic equations.