What Are The Solutions To $(x-4)^2 - 56 = 0$?

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it has numerous applications in various fields such as physics, engineering, and economics. In this article, we will focus on solving the quadratic equation $(x-4)^2 - 56 = 0$, which is a type of quadratic equation that can be solved using various methods.

Understanding the Equation

The given equation is a quadratic equation in the form of $(x-a)^2 + b = 0$, where aa and bb are constants. In this case, a=4a = 4 and b=−56b = -56. To solve this equation, we need to isolate the variable xx.

Method 1: Expanding the Equation

One way to solve this equation is to expand the squared term and then simplify the equation. We can start by expanding the squared term using the formula $(x-a)^2 = x^2 - 2ax + a^2$.

import sympy as sp

x = sp.symbols('x')



equation = (x-4)**2 - 56



expanded_equation = sp.expand(equation)
print(expanded_equation)

This will give us the expanded equation x2−8x+16−56=0x^2 - 8x + 16 - 56 = 0, which can be simplified to x2−8x−40=0x^2 - 8x - 40 = 0.

Method 2: Using the Square Root Method

Another way to solve this equation is to use the square root method. We can start by adding 5656 to both sides of the equation to get $(x-4)^2 = 56$. Then, we can take the square root of both sides to get $x-4 = \pm \sqrt{56}$.

import math


constant = 56



square_root = math.sqrt(constant)
print(square_root)

This will give us the square root of 5656, which is approximately 7.4837.483.

Method 3: Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that the solutions to the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

import math


a = 1
b = -8
c = -40



discriminant = b**2 - 4ac



solution1 = (-b + math.sqrt(discriminant)) / (2a)
solution2 = (-b - math.sqrt(discriminant)) / (2a)


print(solution1)
print(solution2)

This will give us the two solutions to the equation.

Conclusion

In this article, we have discussed three methods for solving the quadratic equation $(x-4)^2 - 56 = 0$. We have used the expanding method, the square root method, and the quadratic formula to find the solutions to the equation. The solutions to the equation are x=4±7.483x = 4 \pm 7.483, which can be simplified to x=11.483x = 11.483 and x=−3.483x = -3.483.

Final Answer

The final answer to the equation $(x-4)^2 - 56 = 0$ is x=11.483x = 11.483 and x=−3.483x = -3.483.

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Solving Quadratic Equations" by Khan Academy
  • [3] "Quadratic Formula" by Wolfram MathWorld

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Introduction

Solving quadratic equations can be a challenging task, especially for those who are new to mathematics. In this article, we will address some of the most frequently asked questions about solving quadratic equations, including the methods used, the types of equations that can be solved, and the tools that can be used to solve them.

Q: What is a quadratic equation?

A: A quadratic equation is a type of polynomial equation that can be written in the form of ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

Q: What are the methods used to solve quadratic equations?

A: There are several methods used to solve quadratic equations, including:

  • Expanding the equation
  • Using the square root method
  • Using the quadratic formula
  • Factoring the equation
  • Graphing the equation

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It states that the solutions to the quadratic equation ax2+bx+c=0ax^2 + bx + c = 0 are given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the difference between the quadratic formula and the square root method?

A: The quadratic formula and the square root method are both used to solve quadratic equations, but they are used in different situations. The quadratic formula is used when the equation is in the form of ax2+bx+c=0ax^2 + bx + c = 0, while the square root method is used when the equation is in the form of (x−a)2+b=0(x-a)^2 + b = 0.

Q: Can quadratic equations be solved using a calculator?

A: Yes, quadratic equations can be solved using a calculator. Many calculators have a built-in quadratic formula function that can be used to solve quadratic equations.

Q: Can quadratic equations be solved using a computer program?

A: Yes, quadratic equations can be solved using a computer program. Many computer programs, such as Python and MATLAB, have built-in functions for solving quadratic equations.

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

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

  • Not following the order of operations
  • Not simplifying the equation before solving it
  • Not checking the solutions to see if they are valid
  • Not using the correct method for solving the equation

Q: How can I practice solving quadratic equations?

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

  • Using online resources, such as Khan Academy and Math Open Reference
  • Working with a tutor or teacher
  • Practicing with sample problems
  • Using a calculator or computer program to solve equations

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

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

  • Physics: Quadratic equations are used to describe 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.

Conclusion

Solving quadratic equations can be a challenging task, but with practice and patience, it can be mastered. By understanding the methods used to solve quadratic equations, the types of equations that can be solved, and the tools that can be used to solve them, you can become proficient in solving quadratic equations and apply them to real-world problems.

Final Answer

The final answer to the question of how to solve quadratic equations is to use a combination of methods, including expanding the equation, using the square root method, using the quadratic formula, factoring the equation, and graphing the equation. With practice and patience, you can become proficient in solving quadratic equations and apply them to real-world problems.

References

  • [1] "Quadratic Equations" by Math Open Reference
  • [2] "Solving Quadratic Equations" by Khan Academy
  • [3] "Quadratic Formula" by Wolfram MathWorld

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