Correct Simplification Of Quadratic Formula For 8x² - 2x = 1

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. Understanding its application and simplification is crucial for success in algebra and beyond. This article will delve into the correct simplification of the quadratic formula when applied to the equation $8x^2 - 2x = 1$, providing a step-by-step guide and clarifying common pitfalls. We will explore the underlying concepts, the formula itself, and the necessary algebraic manipulations to arrive at the correct solution. By the end of this discussion, you'll not only know the answer but also grasp the 'why' behind each step, empowering you to tackle similar problems with confidence. This comprehensive exploration will ensure a solid understanding of the quadratic formula and its applications, making it easier to solve complex mathematical problems. Furthermore, we'll highlight the importance of accuracy in each step and offer strategies for avoiding common errors. This detailed approach aims to equip you with the skills and knowledge necessary to excel in your mathematical pursuits.

Understanding the Quadratic Formula

The quadratic formula is a cornerstone of algebra, providing a universal method for solving equations in the form of $ax^2 + bx + c = 0$. Before diving into the specific problem, let's solidify our understanding of this essential formula. The quadratic formula is given by:

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

Where:

• a is the coefficient of the $x^2$ term
• b is the coefficient of the x term
• c is the constant term

The formula arises from the process of completing the square on the general quadratic equation. It provides two possible solutions for x, represented by the $\pm$ symbol, which indicates both addition and subtraction. The expression under the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant reveals the nature of the roots of the quadratic equation:

• If $b^2 - 4ac > 0$, the equation has two distinct real roots.
• If $b^2 - 4ac = 0$, the equation has one real root (a repeated root).
• If $b^2 - 4ac < 0$, the equation has two complex roots.

A deep understanding of the quadratic formula involves not just memorizing it but also grasping its derivation and the significance of its components. Each variable and operation plays a crucial role in determining the solutions of the equation. By carefully examining the discriminant, we can predict the type of solutions we will encounter before even applying the full formula. This proactive approach enhances our problem-solving skills and helps us avoid potential errors. Furthermore, understanding the formula's origins in completing the square provides a valuable connection between different algebraic techniques. This holistic view of the quadratic formula makes it a powerful tool in our mathematical arsenal.

Setting up the Equation

The given equation is $8x^2 - 2x = 1$. To apply the quadratic formula, we first need to rewrite the equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. This involves moving all terms to one side of the equation, leaving zero on the other side. In our case, we subtract 1 from both sides of the equation:

$8x^2 - 2x - 1 = 0$

Now we can easily identify the coefficients:

• $a = 8$
• $b = -2$
• $c = -1$

Correctly identifying these coefficients is crucial for the accurate application of the quadratic formula. A mistake in this step will lead to incorrect solutions. It's essential to pay close attention to the signs (positive or negative) of each coefficient. A common error is overlooking the negative sign when a term is subtracted. Therefore, double-checking these values before proceeding is a wise practice. This meticulous approach ensures that the subsequent calculations are based on accurate information, maximizing the chances of arriving at the correct answer. Furthermore, a clear understanding of the standard quadratic form helps in recognizing and solving various types of quadratic equations efficiently. By mastering this preliminary step, we lay a solid foundation for successfully applying the quadratic formula.

Applying the Quadratic Formula

Now that we have identified the coefficients $a = 8$, $b = -2$, and $c = -1$, we can substitute these values into the quadratic formula:

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

Substituting the values, we get:

$x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(8)(-1)}}{2(8)}$

This step involves careful substitution, ensuring that each coefficient is placed in the correct position within the formula. It's essential to maintain the correct signs and order of operations to avoid errors. A common mistake is misplacing a coefficient or forgetting a negative sign. Therefore, it's beneficial to write out the formula explicitly with the coefficients before simplifying, ensuring a clear and organized approach. This meticulousness in substitution significantly reduces the likelihood of errors in the subsequent calculations. Furthermore, understanding the structure of the formula helps in recognizing potential errors during the substitution process. By systematically substituting the values, we set the stage for a smooth and accurate simplification process, ultimately leading to the correct solutions of the quadratic equation. This careful approach reinforces the importance of precision in mathematical problem-solving.

Simplifying the Expression

After substituting the values into the quadratic formula, we now need to simplify the expression. Let's break down the simplification step-by-step:

1. Simplify the terms inside the square root:

   • $(-2)^2 = 4$
   • $4(8)(-1) = -32$

   So, $(-2)^2 - 4(8)(-1) = 4 - (-32) = 4 + 32 = 36$

2. Simplify the terms outside the square root:

   • $-(-2) = 2$
   • $2(8) = 16$

Now, substitute these simplified values back into the formula:

$x = \frac{2 \pm \sqrt{36}}{16}$

3. Simplify the square root:

   • $\sqrt{36} = 6$

So, the expression becomes:

$x = \frac{2 \pm 6}{16}$

This stage of simplification is critical as it involves several arithmetic operations. It's imperative to follow the order of operations (PEMDAS/BODMAS) to avoid mistakes. Pay close attention to the signs and ensure accurate calculations at each step. A common error is mishandling the negative signs or miscalculating the square root. Therefore, double-checking each calculation is highly recommended. By breaking down the simplification into smaller, manageable steps, we can minimize the chances of error and maintain clarity in our work. Furthermore, practicing these simplification techniques enhances our arithmetic skills and builds confidence in solving complex expressions. This meticulous approach to simplification is a key element in mastering the quadratic formula and achieving accurate solutions.

Identifying the Correct Option

After simplifying the expression, we have:

$x = \frac{2 \pm 6}{16}$

Comparing this with the given options:

• A. $\frac{2 \pm \sqrt{-28}}{16}$
• B. $\frac{-2 \pm \sqrt{36}}{16}$
• C. $\frac{2 \pm \sqrt{36}}{16}$

We can see that option C, $\frac{2 \pm \sqrt{36}}{16}$, matches our simplified expression. Therefore, option C is the correct answer.

This final step of identifying the correct option reinforces the importance of accurate calculations throughout the entire process. A single error in any of the previous steps could lead to selecting the wrong answer. It's essential to carefully compare the simplified expression with each of the given options, paying attention to every detail. This meticulous comparison helps in verifying the correctness of the solution and avoids careless mistakes. Furthermore, understanding the structure of the options and how they relate to the quadratic formula can aid in quickly identifying the correct answer. By systematically eliminating incorrect options and confirming the match with the simplified expression, we can confidently arrive at the correct solution. This final check serves as a crucial step in ensuring the accuracy and completeness of our problem-solving process.

Common Mistakes to Avoid

When working with the quadratic formula, several common mistakes can occur. Being aware of these pitfalls can help you avoid them:

1. Incorrectly identifying coefficients: A mistake in identifying a, b, or c will lead to an incorrect answer. Always rewrite the equation in the standard form $ax^2 + bx + c = 0$ and double-check the signs.
2. Sign errors: Pay close attention to negative signs, especially when substituting values into the formula and simplifying. For example, $-(-2)$ becomes $+2$, and misinterpreting this can lead to errors.
3. Order of operations: Follow the correct order of operations (PEMDAS/BODMAS) when simplifying the expression, especially the term under the square root.
4. Simplifying the square root: Ensure you simplify the square root correctly. If the number under the square root is not a perfect square, you may need to simplify it by factoring out perfect square factors.
5. Arithmetic errors: Simple arithmetic mistakes can occur during calculations. Double-check each step to ensure accuracy.

By understanding these common mistakes, you can develop strategies to avoid them. One effective strategy is to show your work clearly, writing down each step in detail. This makes it easier to identify and correct any errors. Another helpful technique is to estimate the solutions before applying the quadratic formula, which can help you identify if your final answer is reasonable. Regular practice and a meticulous approach are key to mastering the quadratic formula and avoiding these common pitfalls. This proactive approach to error prevention significantly enhances your problem-solving skills and builds confidence in your mathematical abilities.

Conclusion

In conclusion, the correct simplification of the quadratic formula used to solve $8x^2 - 2x = 1$ is $\frac{2 \pm \sqrt{36}}{16}$, which corresponds to option C. This article has provided a detailed step-by-step explanation of the process, from setting up the equation to simplifying the expression and identifying the correct answer. By understanding the underlying concepts, the formula itself, and the necessary algebraic manipulations, you can confidently tackle similar problems. Remember to pay attention to common mistakes and develop strategies to avoid them. Mastering the quadratic formula is a crucial skill in algebra and will serve you well in more advanced mathematical studies. This comprehensive exploration has aimed to equip you with the knowledge and skills necessary to excel in solving quadratic equations. Regular practice and a meticulous approach will further solidify your understanding and enhance your problem-solving abilities. The quadratic formula is a powerful tool, and with a solid grasp of its application and simplification, you can confidently navigate a wide range of mathematical challenges.