Solving $7x^2 - X = 7$ Using The Quadratic Formula A Step-by-Step Guide

The quadratic formula is an essential tool in algebra for solving quadratic equations, which are equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a ≠ 0$. In this comprehensive guide, we will walk through the process of using the quadratic formula to solve the equation $7x^2 - x = 7$. This article aims to provide a clear, step-by-step explanation that will help you understand and apply the quadratic formula effectively. By the end of this guide, you will be able to confidently solve similar quadratic equations and grasp the underlying principles of this fundamental algebraic technique.

Understanding the Quadratic Formula

The quadratic formula is a powerful tool used to find the solutions (also called roots) of any quadratic equation. A quadratic equation is an equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants and $a$ is not equal to zero. The quadratic formula is given by:

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

This formula provides two possible solutions for $x$, which are obtained by using both the plus and minus signs in the ± symbol. The expression inside the square root, $b^2 - 4ac$, is called the discriminant. The discriminant is very important because its value determines the nature of the solutions:

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

Before we apply the quadratic formula, it is crucial to ensure the quadratic equation is in the standard form $ax^2 + bx + c = 0$. This involves rearranging the terms so that all terms are on one side of the equation, and the other side is equal to zero. Identifying the coefficients $a$, $b$, and $c$ correctly is essential for accurate substitution into the formula. Once these values are determined, substituting them into the quadratic formula and simplifying will lead to the solutions of the equation. Understanding these preliminary steps ensures that the quadratic formula can be applied effectively and the correct solutions can be obtained.

Step 1: Rewrite the Equation in Standard Form

To begin, we need to rewrite the given equation, $7x^2 - x = 7$, 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. Start by subtracting 7 from both sides of the equation:

$7x^2 - x - 7 = 0$

Now, the equation is in the standard form. This is a crucial step because the quadratic formula is designed to work with equations in this format. The standard form allows us to easily identify the coefficients $a$, $b$, and $c$, which are essential for the next steps in applying the formula. Ensuring the equation is correctly rearranged prevents errors in the subsequent calculations and helps in obtaining the correct solutions. This initial rearrangement sets the stage for a successful application of the quadratic formula.

Step 2: Identify the Coefficients a, b, and c

Now that our equation is in the standard form $ax^2 + bx + c = 0$, which is $7x^2 - x - 7 = 0$, we can identify the coefficients $a$, $b$, and $c$. These coefficients are the numerical values that correspond to the terms in the quadratic equation. By comparing the given equation with the standard form, we can easily extract these values:

• The coefficient $a$ is the number multiplying the $x^2$ term. In this case, $a = 7$.
• The coefficient $b$ is the number multiplying the $x$ term. Here, $b = -1$ (note the negative sign).
• The coefficient $c$ is the constant term, which is the number without any $x$ attached. In our equation, $c = -7$.

Correctly identifying these coefficients is a critical step because they are directly used in the quadratic formula. An error in identifying any of these values will lead to incorrect solutions. Therefore, taking a moment to double-check these values ensures the accuracy of the subsequent steps. With the coefficients $a = 7$, $b = -1$, and $c = -7$ clearly identified, we are now ready to substitute these values into the quadratic formula.

Step 3: Apply the Quadratic Formula

With the coefficients $a$, $b$, and $c$ identified, we can now apply the quadratic formula:

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

Substitute the values $a = 7$, $b = -1$, and $c = -7$ into the formula:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-7)}}{2(7)}$

This substitution is a crucial step, and it is important to perform it accurately. Be particularly careful with the signs, as errors in the signs can significantly affect the outcome. After substituting, the next step is to simplify the expression. We begin by simplifying the terms inside the square root and the denominator. Careful attention to detail during this simplification process is essential to ensure the correct solution. By correctly substituting the values and setting up the expression, we pave the way for a successful simplification and the final calculation of the roots of the equation.

Step 4: Simplify the Expression

After substituting the values into the quadratic formula, we have:

$x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(7)(-7)}}{2(7)}$

Now, let's simplify the expression step by step. First, simplify the terms inside the square root:

$(-1)^2 = 1$

$4(7)(-7) = -196$

So, the expression inside the square root becomes:

$1 - (-196) = 1 + 196 = 197$

Next, simplify the denominator:

$2(7) = 14$

Now, the entire expression simplifies to:

$x = \frac{1 \pm \sqrt{197}}{14}$

This simplified form is much easier to work with and brings us closer to the final solution. Each step in the simplification process is critical to ensure accuracy. By carefully simplifying the terms inside the square root and the denominator, we reduce the complexity of the equation and make it easier to handle. The simplified expression now clearly shows the two possible solutions for $x$, which we will finalize in the next step.

Step 5: State the Solutions

After simplifying the expression, we have:

$x = \frac{1 \pm \sqrt{197}}{14}$

This expression represents two solutions for $x$, one with the plus sign and one with the minus sign. So, the solutions are:

$x_1 = \frac{1 + \sqrt{197}}{14}$

$x_2 = \frac{1 - \sqrt{197}}{14}$

These are the exact solutions to the quadratic equation $7x^2 - x = 7$. We have successfully used the quadratic formula to find the values of $x$ that satisfy the equation. Stating the solutions explicitly provides a clear and final answer to the problem. Each solution represents a point where the quadratic function intersects the x-axis. By clearly stating these solutions, we complete the process of solving the quadratic equation and provide a comprehensive answer that can be easily understood and applied.

Conclusion

In this guide, we have demonstrated how to solve the quadratic equation $7x^2 - x = 7$ using the quadratic formula. By following the step-by-step process, we first rewrote the equation in standard form, identified the coefficients $a$, $b$, and $c$, substituted these values into the quadratic formula, simplified the expression, and finally stated the two solutions for $x$. The solutions we found are:

$x_1 = \frac{1 + \sqrt{197}}{14}$

$x_2 = \frac{1 - \sqrt{197}}{14}$

Understanding and applying the quadratic formula is a fundamental skill in algebra. This method allows you to solve any quadratic equation, regardless of whether it can be factored easily. The key to success is to follow each step carefully and pay attention to detail, especially when substituting values and simplifying expressions. Mastering this technique will greatly enhance your ability to tackle more complex algebraic problems and provide a solid foundation for further mathematical studies. By practicing and understanding the principles outlined in this guide, you can confidently approach and solve a wide range of quadratic equations.

Therefore, the correct answer is B. $\frac{1 \pm \sqrt{197}}{14}$.