Select The Correct Answer.Solve The Quadratic Equation Given Below:${(12x - 27)^2 = 256}$A. { X = -\frac{43}{12}; -\frac{11}{12}$}$ B. { X = -\frac{43}{12}; \frac{11}{12}$}$ C. [$x = \frac{43}{12};

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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, and we will provide a step-by-step guide on how to arrive at the correct solution.

The Quadratic Equation

The quadratic equation given in the problem is:

(12x−27)2=256{(12x - 27)^2 = 256}

This equation can be solved using various methods, including factoring, the quadratic formula, and completing the square. In this article, we will use the method of factoring to solve the equation.

Step 1: Expand the Equation

The first step in solving the equation is to expand the left-hand side using the formula (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2. In this case, a=12xa = 12x and b=27b = 27. Therefore, we have:

(12x−27)2=(12x)2−2(12x)(27)+272{(12x - 27)^2 = (12x)^2 - 2(12x)(27) + 27^2}

Simplifying the equation, we get:

144x2−108xy+729=256{144x^2 - 108xy + 729 = 256}

Step 2: Rearrange the Equation

The next step is to rearrange the equation to get a quadratic equation in the form ax2+bx+c=0ax^2 + bx + c = 0. Subtracting 256 from both sides, we get:

144x2−108xy−527=0{144x^2 - 108xy - 527 = 0}

Step 3: Factor the Equation

The equation can be factored as follows:

(12x−43)(12x+12)=0{(12x - 43)(12x + 12) = 0}

Step 4: Solve for x

To solve for x, we need to set each factor equal to zero and solve for x. Therefore, we have:

12x−43=0or12x+12=0{12x - 43 = 0 \quad \text{or} \quad 12x + 12 = 0}

Solving for x, we get:

x=4312orx=−112{x = \frac{43}{12} \quad \text{or} \quad x = -\frac{1}{12}}

Conclusion

In conclusion, the correct solution to the quadratic equation is:

x=4312orx=−112{x = \frac{43}{12} \quad \text{or} \quad x = -\frac{1}{12}}

Therefore, the correct answer is:

A. {x = -\frac{43}{12}; -\frac{11}{12}$}$

Note that the other options are incorrect, and the correct solution is a combination of the two values of x.

Discussion

The solution to the quadratic equation involves several steps, including expanding the equation, rearranging it, factoring it, and solving for x. The key to solving the equation is to recognize that it can be factored as a product of two binomials. Once the equation is factored, it is easy to solve for x by setting each factor equal to zero and solving for x.

Tips and Tricks

  • When solving quadratic equations, it is essential to recognize that the equation can be factored as a product of two binomials.
  • To factor the equation, look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term.
  • When solving for x, make sure to check for extraneous solutions by plugging the value of x back into the original equation.

Common Mistakes

  • One common mistake when solving quadratic equations is to forget to check for extraneous solutions.
  • Another common mistake is to incorrectly factor the equation, which can lead to incorrect solutions.

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 bridges, buildings, and other structures.
  • Economics: Quadratic equations are used to model the behavior of economic systems.

Conclusion

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 provide a Q&A guide to help you understand and solve quadratic equations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means it has a highest power of two. It is typically written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, the quadratic formula, and completing the square. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations. It is given by:

x=−b±b2−4ac2a{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 aa, bb, and cc into the formula. Then, simplify the expression and solve for xx.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula and factoring are two different methods for solving quadratic equations. The quadratic formula is a formula that can be used to solve any quadratic equation, while factoring is a method that involves finding two binomials whose product is equal to the original equation.

Q: When should I use the quadratic formula?

A: You should use the quadratic formula when the equation cannot be factored easily, or when you are dealing with a complex equation.

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

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

  • Forgetting to check for extraneous solutions
  • Incorrectly factoring the equation
  • Not simplifying the expression correctly
  • Not plugging the value of xx back into the original equation to check for extraneous solutions

Q: How do I check for extraneous solutions?

A: To check for extraneous solutions, plug the value of xx back into the original equation and simplify. If the equation is true, then the value of xx is a valid solution. If the equation is false, then the value of xx is an extraneous solution.

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 bridges, buildings, and other structures.
  • Economics: Quadratic equations are used to model the behavior of economic systems.

Q: How do I choose the correct method for solving a quadratic equation?

A: To choose the correct method for solving a quadratic equation, consider the following factors:

  • The complexity of the equation
  • The ease of factoring the equation
  • The need for a specific solution

Conclusion

In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can arrive at the correct solution to the quadratic equation. Remember to recognize that the equation can be factored as a product of two binomials, and to check for extraneous solutions by plugging the value of xx back into the original equation.

Additional Resources

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Practice Problems

  • Solve the quadratic equation x2+5x+6=0x^2 + 5x + 6 = 0 using factoring.
  • Solve the quadratic equation x2−7x+12=0x^2 - 7x + 12 = 0 using the quadratic formula.
  • Solve the quadratic equation x2+2x−15=0x^2 + 2x - 15 = 0 using completing the square.

Answer Key

  • x=−2x = -2 or x=−3x = -3
  • x=3x = 3 or x=4x = 4
  • x=−3x = -3 or x=5x = 5