Select The Correct Answer.Solve The Equation:$3x^2 + 24x - 24 = 0$A. $x = 2 \pm 4 \sqrt{6}$ B. $x = -4 \pm 2 \sqrt{6}$ C. $x = 4 \pm 2 \sqrt{6}$ D. $x = -2 \pm 4 \sqrt{6}$

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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: 3x2+24x−24=03x^2 + 24x - 24 = 0. We will break down the solution step by step, using the quadratic formula and other mathematical techniques to arrive at the correct 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, xx) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants, and xx is the variable. In our equation, a=3a = 3, b=24b = 24, and c=−24c = -24.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula will give us two solutions for the equation, which we will use to determine the correct answer.

Applying the Quadratic Formula

Now, let's apply the quadratic formula to our equation:

x=−24±(24)2−4(3)(−24)2(3)x = \frac{-24 \pm \sqrt{(24)^2 - 4(3)(-24)}}{2(3)}

Simplifying the expression inside the square root, we get:

x=−24±576+2886x = \frac{-24 \pm \sqrt{576 + 288}}{6}

x=−24±8646x = \frac{-24 \pm \sqrt{864}}{6}

x=−24±2466x = \frac{-24 \pm 24\sqrt{6}}{6}

Simplifying the Solutions

Now, let's simplify the solutions by dividing both the numerator and the denominator by 6:

x=−246±2466x = \frac{-24}{6} \pm \frac{24\sqrt{6}}{6}

x=−4±46x = -4 \pm 4\sqrt{6}

Conclusion

We have successfully solved the quadratic equation 3x2+24x−24=03x^2 + 24x - 24 = 0 using the quadratic formula. The solutions are:

x=−4±46x = -4 \pm 4\sqrt{6}

This matches option A in the given choices.

Discussion

The quadratic formula is a powerful tool for solving quadratic equations. It is a fundamental concept in mathematics and has numerous applications in science, engineering, and other fields. In this article, we have demonstrated how to apply the quadratic formula to solve a specific quadratic equation. We have also shown how to simplify the solutions and arrive at the correct answer.

Final Answer

The correct answer is:

  • A. x=2±46x = 2 \pm 4 \sqrt{6}

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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 our previous article, we solved a specific quadratic equation using the quadratic formula. In this article, we will provide a comprehensive Q&A guide to help you understand quadratic equations and how to solve them.

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, xx) is two. The general form of a quadratic equation is:

ax2+bx+c=0ax^2 + bx + c = 0

where aa, bb, and cc are constants, and xx is the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula will give us two solutions for the equation, which we will use to determine the correct answer.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to substitute the values of aa, bb, and cc into the formula. Then, simplify the expression inside the square root and solve for xx.

Q: What is the difference between the two solutions given by the quadratic formula?

A: The two solutions given by the quadratic formula are:

x=−b+b2−4ac2ax = \frac{-b + \sqrt{b^2 - 4ac}}{2a}

x=−b−b2−4ac2ax = \frac{-b - \sqrt{b^2 - 4ac}}{2a}

These two solutions are called the positive and negative solutions, respectively.

Q: How do I determine which solution is correct?

A: To determine which solution is correct, you need to check if the solution satisfies the original equation. You can do this by substituting the solution back into the original equation and checking if it is true.

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

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

  • Not simplifying the expression inside the square root
  • Not checking if the solution satisfies the original equation
  • Not using the correct values of aa, bb, and cc

Q: How do I use the quadratic formula to solve a quadratic equation with complex solutions?

A: To use the quadratic formula to solve a quadratic equation with complex solutions, you need to simplify the expression inside the square root and then use the complex conjugate to find the solutions.

Q: What is the significance of quadratic equations in real-life applications?

A: Quadratic equations have numerous applications in science, engineering, and other fields. Some examples include:

  • 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.
  • Computer Science: Quadratic equations are used in algorithms and data structures to solve problems efficiently.

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 have provided a comprehensive Q&A guide to help you understand quadratic equations and how to solve them. We have also discussed some common mistakes to avoid and the significance of quadratic equations in real-life applications.

Final Answer

The correct answer is:

  • A. x=2±46x = 2 \pm 4 \sqrt{6}