Solving 7x² - X = 7 Using The Quadratic Formula

The quadratic formula is a powerful tool in algebra for finding the solutions, also known as 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. This comprehensive guide will walk you through the process of using the quadratic formula to solve the specific equation $7x^2 - x = 7$. We'll break down each step, ensuring you understand not only how to apply the formula but also why it works. By the end of this article, you'll be well-equipped to tackle similar problems and have a deeper understanding of quadratic equations.

The quadratic formula is derived from the method of completing the square and is expressed as:

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

Where:

• $x$ represents the solutions (or roots) of the quadratic equation.
• $a$, $b$, and $c$ are the coefficients from the quadratic equation in the standard form $ax^2 + bx + c = 0$.
• The $\pm$ symbol indicates that there are typically two solutions: one where you add the square root term and one where you subtract it.

Before diving into our specific equation, it's crucial to understand each component of the formula and how it relates to the quadratic equation. The term inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant provides valuable information about the nature of the roots:

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

Understanding the discriminant can help you anticipate the type of solutions you'll find, making it a useful tool in problem-solving.

To use the quadratic formula effectively, the first step is to rewrite the given equation in the standard form $ax^2 + bx + c = 0$. Our initial equation is $7x^2 - x = 7$. To bring it to the standard form, we need to subtract 7 from both sides of the equation:

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

Now, we can clearly identify the coefficients:

• $a = 7$
• $b = -1$
• $c = -7$

Ensuring the equation is in standard form is a critical step because the coefficients $a$, $b$, and $c$ are directly used in the quadratic formula. Misidentifying these values will lead to incorrect solutions. Double-checking this step can save you from potential errors later on.

Now that we have the coefficients, we can substitute them into the quadratic formula:

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

Plugging in $a = 7$, $b = -1$, and $c = -7$, we get:

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

This substitution is a crucial step, and it's important to be meticulous to avoid errors. Pay close attention to signs and ensure that each value is placed correctly in the formula. A common mistake is mishandling negative signs, so take your time and double-check your work.

Next, we simplify the expression step by step. First, let's simplify the terms inside the square root and the denominator:

$x = \frac{1 \pm \sqrt{1 - (-196)}}{14}$

$x = \frac{1 \pm \sqrt{1 + 196}}{14}$

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

Simplifying the expression involves basic arithmetic operations, but it's essential to proceed carefully. Breaking down the simplification into smaller steps can help reduce the chances of making mistakes. In this case, we've simplified the expression under the square root and the denominator, setting us up for the final step.

Now we have the simplified expression:

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

The $\pm$ symbol indicates that we have two solutions. We can write them separately:

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

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

These are the two solutions to the quadratic equation $7x^2 - x = 7$. The solutions are real numbers because the discriminant (197) is positive. We have successfully used the quadratic formula to find the values of $x$ that satisfy the equation.

In this comprehensive guide, we've demonstrated how to use the quadratic formula to solve the equation $7x^2 - x = 7$. We started by understanding the formula itself and its components, emphasizing the importance of the discriminant. We then meticulously worked through the steps:

1. Rewriting the equation in standard form.
2. Substituting the values into the quadratic formula.
3. Simplifying the expression.
4. Identifying the solutions.

By following these steps, we found the two solutions to the equation:

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

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

Understanding and applying the quadratic formula is a fundamental skill in algebra. With practice, you'll become more comfortable and confident in using it to solve a wide range of quadratic equations. Remember to always double-check your work and take your time to avoid common errors.

The final answer is B. $\frac{1 \pm \sqrt{197}}{14}$