Solve For \[$ P \$\].\[$(p-4)^2 = 49\$\]
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-4)^2 = 49, to find the value of p. We will break down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer.
Understanding 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. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.
The Given Equation
The given equation is (p-4)^2 = 49. This is a quadratic equation in the form of (x-a)^2 = b, where x is the variable, a is a constant, and b is a constant. To solve for p, we need to isolate p on one side of the equation.
Step 1: Expand the Left Side of the Equation
To expand the left side of the equation, we need to apply the formula (x-a)^2 = x^2 - 2ax + a^2. In this case, x = p and a = 4. Therefore, we can expand the left side of the equation as follows:
(p-4)^2 = p^2 - 2(4)p + 4^2
Step 2: Simplify the Equation
Now that we have expanded the left side of the equation, we can simplify it by combining like terms. The equation becomes:
p^2 - 8p + 16 = 49
Step 3: Rearrange the Equation
To isolate p on one side of the equation, we need to move all the terms to one side of the equation. We can do this by subtracting 49 from both sides of the equation:
p^2 - 8p + 16 - 49 = 0
This simplifies to:
p^2 - 8p - 33 = 0
Step 4: Solve for p
Now that we have a quadratic equation in the form of ax^2 + bx + c = 0, we can use the quadratic formula to solve for p. The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -8, and c = -33. Plugging these values into the quadratic formula, we get:
p = (8 ± √((-8)^2 - 4(1)(-33))) / 2(1)
Step 5: Simplify the Expression
Now that we have the quadratic formula, we can simplify the expression under the square root. The expression becomes:
p = (8 ± √(64 + 132)) / 2
This simplifies to:
p = (8 ± √196) / 2
Step 6: Solve for p
Now that we have simplified the expression under the square root, we can solve for p. The expression becomes:
p = (8 ± 14) / 2
This gives us two possible values for p:
p = (8 + 14) / 2 = 22 / 2 = 11
p = (8 - 14) / 2 = -6 / 2 = -3
Conclusion
In this article, we solved a quadratic equation, (p-4)^2 = 49, to find the value of p. We broke down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer. The two possible values for p are 11 and -3.
Final Answer
The final answer is:
Introduction
In our previous article, we solved a quadratic equation, (p-4)^2 = 49, to find the value of p. We broke down the solution into manageable steps, using algebraic manipulations and mathematical concepts to arrive at the final answer. In this article, we will answer some frequently asked questions about quadratic equations and solving for p.
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.
Q: How do I know if a quadratic equation can be solved using factoring?
A: To determine if a quadratic equation can be solved using factoring, you need to check if the equation can be written in the form of (x-a)(x-b) = 0, where a and b are constants. If the equation can be factored, then it can be solved using factoring.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. The formula is:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula to solve a quadratic equation?
A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of a, b, and c into the formula. The formula will give you two possible values for x.
Q: What are the two possible values for p in the equation (p-4)^2 = 49?
A: The two possible values for p in the equation (p-4)^2 = 49 are 11 and -3.
Q: How do I know which value of p is correct?
A: To determine which value of p is correct, you need to plug the values back into the original equation and check if the equation is true. If the equation is true, then the value of p is correct.
Q: Can I use a calculator to solve a quadratic equation?
A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to solve 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 if the equation can be factored
- Not using the correct values of a, b, and c in the quadratic formula
- Not plugging the values back into the original equation to check if the equation is true
- Not using a calculator to check the solution
Conclusion
In this article, we answered some frequently asked questions about quadratic equations and solving for p. We covered topics such as factoring, the quadratic formula, and common mistakes to avoid. We hope that this article has been helpful in understanding quadratic equations and solving for p.
Final Answer
The final answer is:
p = 11 or p = -3