Solve For { P $}$ In The Equation:${ P^2 + 5p - 84 = 0 }$

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 $p^2 + 5p - 84 = 0$ to find the value of $p$. We will use the quadratic formula and factorization methods to solve this equation.

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, and $x$ is the variable. In our equation, $p$ is the variable, and the coefficients are $a = 1$, $b = 5$, and $c = -84$.

The Quadratic Formula

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

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

In our equation, $a = 1$, $b = 5$, and $c = -84$. Plugging these values into the quadratic formula, we get:

$$p = \frac{-5 \pm \sqrt{5^2 - 4(1)(-84)}}{2(1)}$$

Simplifying the Quadratic Formula

To simplify the quadratic formula, we need to calculate the value inside the square root. This involves calculating the discriminant, which is the expression under the square root.

$$b^2 - 4ac = 5^2 - 4(1)(-84) = 25 + 336 = 361$$

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

$$p = \frac{-5 \pm \sqrt{361}}{2}$$

Solving for p

The square root of 361 is 19, so we can simplify the equation further:

$$p = \frac{-5 \pm 19}{2}$$

This gives us two possible solutions for $p$:

$$p = \frac{-5 + 19}{2} = \frac{14}{2} = 7$$

$$p = \frac{-5 - 19}{2} = \frac{-24}{2} = -12$$

Checking the Solutions

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

$$p^2 + 5p - 84 = 0$$

For $p = 7$, we get:

$$(7)^2 + 5(7) - 84 = 49 + 35 - 84 = 0$$

This confirms that $p = 7$ is a valid solution.

For $p = -12$, we get:

$$( -12)^2 + 5(-12) - 84 = 144 - 60 - 84 = 0$$

This confirms that $p = -12$ is also a valid solution.

Conclusion

In this article, we solved a quadratic equation of the form $p^2 + 5p - 84 = 0$ to find the value of $p$. We used the quadratic formula and factorization methods to solve this equation. We found two possible solutions for $p$, which were $p = 7$ and $p = -12$. We checked our solutions by plugging them back into the original equation, and both solutions were confirmed to be 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 $ax^2 + bx + c = 0$.
• Factorization methods can also be used to solve quadratic equations, but they may not always be possible.

Real-World Applications

Quadratic equations have numerous 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.

Final Thoughts

$$$$$$

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.

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 methods to solve a quadratic equation, including:

• Factoring: This involves expressing the quadratic equation as a product of two binomials.
• Quadratic formula: This involves using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to solve the equation.
• Graphing: This involves graphing the quadratic equation on a coordinate plane and finding the x-intercepts.

Q: What is the quadratic formula?

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

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

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. Then, you need to simplify the expression under the square root and solve for $x$.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It 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: How do I check my solutions?

A: To check your solutions, you need to plug them back into the original equation and verify that they satisfy the equation.

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 your solutions
• Not simplifying the expression under the square root
• Not using the correct formula
• Not following the correct order of operations

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

A: Quadratic equations have numerous 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 can I practice solving quadratic equations?

A: There are several ways to practice solving quadratic equations, including:

• Using online resources, such as Khan Academy or Mathway
• Working with a tutor or teacher
• Practicing with sample problems
• Using real-world applications to solve quadratic equations

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 answered some of the most frequently asked questions about quadratic equations. We hope that this article has been helpful in providing you with a better understanding of quadratic equations and how to solve them.