Solving For X Using The Quadratic Formula When A Equals 7 B Equals 6 And C Equals -8

In this article, we will walk through the process of solving for the variable x using the quadratic formula. The quadratic formula is a fundamental tool in algebra for finding the roots, or solutions, of a quadratic equation. A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants, and a is not equal to zero. The quadratic formula provides a direct method to find the values of x that satisfy this equation.

The Quadratic Formula

The quadratic formula is given by:

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

This formula tells us that there are generally two solutions for x, which are obtained by using the plus (+) and minus (-) signs in the formula. The expression inside the square root, b² - 4ac, is called the discriminant. The discriminant provides valuable information about the nature of the roots: if it's positive, there are two distinct real roots; if it's zero, there is exactly one real root (a repeated root); and if it's negative, there are two complex roots.

Problem Statement

We are given the quadratic formula and specific values for the constants a, b, and c:

• $x = \frac{-b + \sqrt{b^2 - 4ac}}{2a}$
• Given: $a = 7, b = 6$, and $c = -8$

Our goal is to calculate the value of x using the provided formula and the given values. We will substitute the values of a, b, and c into the formula and simplify the expression to find the value of x. We will round the final answer to two decimal places as necessary.

Step-by-Step Solution

Let's break down the solution into manageable steps.

Step 1: Substitute the Given Values

First, we substitute the given values of a, b, and c into the quadratic formula:

$x = \frac{-6 + \sqrt{6^2 - 4(7)(-8)}}{2(7)}$

This step involves simply replacing the variables a, b, and c with their corresponding numerical values. It's crucial to ensure that the substitution is done correctly to avoid errors in the subsequent steps.

Step 2: Simplify the Expression

Next, we simplify the expression inside the square root and the denominator:

$x = \frac{-6 + \sqrt{36 + 224}}{14}$

Here, we've calculated 6² as 36 and -4 * 7 * -8 as 224. We also multiplied 2 by 7 to get 14 in the denominator. Simplifying the expression step by step helps to minimize the chances of making mistakes.

Step 3: Continue Simplifying

Now, we add the numbers inside the square root:

$x = \frac{-6 + \sqrt{260}}{14}$

We've added 36 and 224 to get 260. The expression is now further simplified, bringing us closer to the final answer.

Step 4: Calculate the Square Root

We calculate the square root of 260. The square root of 260 is not a whole number, so we will approximate it to two decimal places:

$\sqrt{260} \approx 16.12$

Therefore,

$x = \frac{-6 + 16.12}{14}$

Approximating the square root to two decimal places ensures that our final answer is also accurate to two decimal places, as required by the problem statement.

Step 5: Perform the Addition

Next, we add -6 and 16.12:

$x = \frac{10.12}{14}$

This step involves a simple addition operation, which is straightforward to perform.

Step 6: Divide to Find x

Finally, we divide 10.12 by 14 to find the value of x:

$x \approx 0.72$

We have now calculated the value of x by performing the division. The result is approximately 0.72.

Final Answer

Therefore, the value of x, rounded to two decimal places, is approximately 0.72.

$x \approx 0.72$

This is the final answer to the problem. We have successfully used the quadratic formula and the given information to solve for x.

Understanding the Solution

In summary, we used the quadratic formula to solve for x. The steps we followed included substituting the given values of a, b, and c, simplifying the expression step by step, calculating the square root, and performing the necessary arithmetic operations. The quadratic formula is a powerful tool for solving quadratic equations, and understanding how to use it is essential in algebra.

This problem illustrates a practical application of the quadratic formula. By following a systematic approach, we can break down complex problems into simpler steps and arrive at the correct solution. Understanding the underlying principles and practicing problem-solving techniques are key to mastering algebra and other areas of mathematics.

Conclusion

In this article, we have demonstrated how to use the quadratic formula to solve for x given specific values for a, b, and c. We followed a step-by-step approach, which included substituting the given values, simplifying the expression, calculating the square root, and performing the final division. The final answer, rounded to two decimal places, is approximately 0.72. This example showcases the importance of the quadratic formula in solving quadratic equations and highlights the significance of a systematic problem-solving approach.

By understanding the quadratic formula and practicing its application, students can build a strong foundation in algebra and tackle more complex mathematical problems with confidence. This article serves as a comprehensive guide to solving for x using the quadratic formula, providing readers with the knowledge and skills needed to succeed in their mathematical endeavors.