Solving 11x^2 - 4x = 1 Using The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form $ax^2 + bx + c = 0$. In this article, we will delve into the process of using the quadratic formula to find the values of $x$ that satisfy the equation $11x^2 - 4x = 1$. We will break down each step, ensuring a clear understanding of the method and its application. This comprehensive guide will not only provide the solution but also explain the underlying concepts, making it a valuable resource for students and anyone looking to enhance their algebra skills. Let's embark on this mathematical journey together and master the art of solving quadratic equations!

Understanding the Quadratic Formula

Before we dive into the specifics of the equation $11x^2 - 4x = 1$, it's crucial to understand the quadratic formula itself. The quadratic formula is derived from the method of completing the square and provides a direct way to find the solutions (also called roots) of any quadratic equation. The formula is given by:

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

Where a, b, and c are the coefficients of the quadratic equation in the standard form $ax^2 + bx + c = 0$. The expression inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant plays a significant role in determining 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 one real root (a repeated root).
• If $b^2 - 4ac < 0$, the equation has two complex roots.

Understanding these fundamental concepts is essential for effectively applying the quadratic formula and interpreting the solutions obtained. Now that we have a solid grasp of the formula, let's move on to applying it to our specific equation.

Transforming the Equation into Standard Form

The first step in using the quadratic formula is to ensure that the given equation is in the standard quadratic form, which is $ax^2 + bx + c = 0$. Our equation is $11x^2 - 4x = 1$. To transform it into standard form, we need to subtract 1 from both sides of the equation:

$11x^2 - 4x - 1 = 0$

Now, we can clearly identify the coefficients a, b, and c:

• $a = 11$
• $b = -4$
• $c = -1$

Having identified these coefficients, we are now ready to substitute them into the quadratic formula and proceed with solving for x. This step is crucial as it sets the stage for the accurate application of the formula and the subsequent determination of the roots of the equation. It is vital to ensure that the coefficients are correctly identified to avoid errors in the final solution.

Applying the Quadratic Formula

With the coefficients identified ($a = 11$, $b = -4$, and $c = -1$), we can now substitute these values into the quadratic formula:

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

Substituting the values, we get:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(11)(-1)}}{2(11)}$

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

$(-4)^2 = 16$

$4(11)(-1) = -44$

So, the expression inside the square root becomes:

$16 - (-44) = 16 + 44 = 60$

Now, we can rewrite the equation as:

$x = \frac{4 \pm \sqrt{60}}{22}$

This is a significant step forward. We have successfully substituted the coefficients into the quadratic formula and simplified the expression under the square root. The next step involves further simplifying the square root and reducing the fraction to obtain the final solutions for x. Let's proceed with this simplification in the following section.

Simplifying the Solution

We have reached the point where we have the equation:

$x = \frac{4 \pm \sqrt{60}}{22}$

To simplify this further, we need to simplify the square root. We can factor 60 as $4 imes 15$. Since 4 is a perfect square, we can take its square root:

$\sqrt{60} = \sqrt{4 imes 15} = \sqrt{4} imes \sqrt{15} = 2\sqrt{15}$

Now, substitute this back into the equation:

$x = \frac{4 \pm 2\sqrt{15}}{22}$

We can now factor out a 2 from the numerator:

$x = \frac{2(2 \pm \sqrt{15})}{22}$

And then simplify the fraction by dividing both the numerator and the denominator by 2:

$x = \frac{2 \pm \sqrt{15}}{11}$

This simplified form gives us the two solutions for x:

$x = \frac{2 + \sqrt{15}}{11}$ and $x = \frac{2 - \sqrt{15}}{11}$

We have successfully simplified the solution obtained from the quadratic formula. This final simplified form provides the exact values of x that satisfy the original equation. The next step is to express these solutions in the format provided in the answer choices and select the correct option.

Identifying the Correct Answer

Our simplified solutions are:

$x = \frac{2 + \sqrt{15}}{11}$ and $x = \frac{2 - \sqrt{15}}{11}$

We can rewrite these as:

$x = \frac{2}{11} + \frac{\sqrt{15}}{11}$ and $x = \frac{2}{11} - \frac{\sqrt{15}}{11}$

Combining these, we get:

$x = \frac{2}{11} \pm \frac{\sqrt{15}}{11}$

Now, let's compare this to the given options:

A. $\frac{2}{11} \pm \frac{\sqrt{15}}{11}$ B. $\frac{2}{11} \pm \frac{2 \sqrt{15}}{11}$ C. $\frac{2}{11} \pm \frac{\sqrt{7}}{11}$ D. $\frac{2}{11} \pm \frac{\sqrt{7}}{11}$

Clearly, option A matches our solution.

Therefore, the correct answer is:

A. $\frac{2}{11} \pm \frac{\sqrt{15}}{11}$

We have successfully identified the correct answer by systematically applying the quadratic formula, simplifying the resulting expression, and comparing our solution with the given options. This step-by-step approach ensures accuracy and clarity in the problem-solving process. The final answer, option A, represents the values of x that satisfy the original quadratic equation.

Conclusion

In this article, we have thoroughly explored the process of using the quadratic formula to solve the equation $11x^2 - 4x = 1$. We started by understanding the quadratic formula and its components, then transformed the given equation into standard form. We carefully substituted the coefficients into the formula, simplified the resulting expression, and ultimately arrived at the solutions:

$x = \frac{2}{11} \pm \frac{\sqrt{15}}{11}$

This step-by-step approach not only provides the correct answer but also enhances understanding of the underlying mathematical concepts. The quadratic formula is a fundamental tool in algebra, and mastering its application is crucial for solving a wide range of problems. By breaking down the process into manageable steps, we have demonstrated how to effectively use this formula to find the roots of a quadratic equation. This guide serves as a valuable resource for students and anyone seeking to improve their problem-solving skills in mathematics. Remember, practice is key to mastering these concepts, so continue to apply the quadratic formula to various equations to solidify your understanding.