Solving $x$ In The Equation $x^2 - 12x + 59 = 0$

Solving quadratic equations is a fundamental skill in algebra. When faced with an equation in the form $ax^2 + bx + c = 0$, we have several methods at our disposal, including factoring, completing the square, and the quadratic formula. This article delves into solving the specific quadratic equation $x^2 - 12x + 59 = 0$ using the quadratic formula, which is a reliable and versatile approach applicable to all quadratic equations. Understanding how to apply the quadratic formula not only helps in finding the solutions to this particular equation but also provides a solid foundation for tackling more complex algebraic problems. We will break down each step, from identifying the coefficients to simplifying the result, ensuring a clear and comprehensive understanding of the solution process. So, let's embark on this journey to find the values of $x$ that satisfy the given equation.

Understanding the Quadratic Formula

Before diving into the specifics of the given equation, let's recap the quadratic formula. The quadratic formula is a powerful tool used to find the roots (or solutions) of any quadratic equation in the standard form of $ax^2 + bx + c = 0$. The formula is expressed as:

x = rac{-b loat{\pm} loat{\sqrt{b^2 - 4ac}}}{2a}

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic equation, where $a$ is the coefficient of the $x^2$ term, $b$ is the coefficient of the $x$ term, and $c$ is the constant term. The expression under the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has exactly one real root (a repeated root). If it is negative, the equation has two complex roots. Understanding these nuances helps in interpreting the solutions we obtain. The quadratic formula provides a systematic way to solve for $x$, ensuring that we don't miss any possible solutions. It is particularly useful when factoring is not straightforward or when the roots are not rational numbers. By mastering the quadratic formula, you can confidently tackle a wide range of quadratic equations, regardless of their complexity. This formula is a cornerstone of algebra and is essential for further studies in mathematics and related fields.

Identifying Coefficients in $x^2 - 12x + 59 = 0$

The first step in applying the quadratic formula to the equation $x^2 - 12x + 59 = 0$ is to correctly identify the coefficients $a$, $b$, and $c$. These coefficients are the numerical values that correspond to the terms in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$. In our equation, the coefficient $a$ is the number multiplying the $x^2$ term. Since there is no explicit number written before $x^2$, it is understood that the coefficient is 1. Therefore, $a = 1$. The coefficient $b$ is the number multiplying the $x$ term. In this case, $b = -12$. It is crucial to include the negative sign, as it significantly impacts the outcome of the quadratic formula. Lastly, the coefficient $c$ is the constant term, which is the number without any $x$ attached. In our equation, $c = 59$. Correctly identifying these coefficients is paramount because they are the building blocks for the quadratic formula. A mistake in identifying any of these coefficients will lead to incorrect solutions. Once we have accurately identified $a = 1$, $b = -12$, and $c = 59$, we can confidently proceed to substitute these values into the quadratic formula. This careful approach ensures that we are setting the stage for a successful and accurate solution. The precision in this initial step is key to unlocking the correct answers and demonstrating a solid understanding of the quadratic formula.

Applying the Quadratic Formula

Now that we have identified the coefficients $a = 1$, $b = -12$, and $c = 59$, we can substitute these values into the quadratic formula:

x = rac{-b loat{\pm} loat{\sqrt{b^2 - 4ac}}}{2a}

Plugging in the values, we get:

x = rac{-(-12) loat{\pm} loat{\sqrt{(-12)^2 - 4(1)(59)}}}{2(1)}

This substitution is a critical step, and it's essential to be meticulous to avoid errors. The negative signs, especially, require careful attention. Once the values are correctly substituted, the next step is to simplify the expression. We start by simplifying the terms inside the square root and the terms outside the square root separately. First, let's simplify the numerator. The term $-(-12)$ becomes $+12$. Next, we simplify the expression under the square root: $(-12)^2$ is $144$, and $4(1)(59)$ is $236$. So, the expression under the square root becomes $144 - 236$. In the denominator, $2(1)$ simplifies to $2$. Thus, our equation now looks like this:

x = rac{12 loat{\pm} loat{\sqrt{144 - 236}}}{2}

By carefully substituting and simplifying, we are methodically moving towards the solution. This step-by-step approach minimizes the chances of making mistakes and ensures a clear and understandable solution process. The next phase involves further simplifying the expression under the square root and proceeding with the calculations.

Simplifying the Expression

Continuing from the previous step, we now need to simplify the expression under the square root. We have $144 - 236$, which results in $-92$. So, the equation becomes:

x = rac{12 loat{\pm} loat{\sqrt{-92}}}{2}

Here, we encounter a negative number under the square root, which indicates that the solutions will be complex numbers. Recall that the square root of a negative number involves the imaginary unit, denoted as $i$, where i = loat{\sqrt{-1}}. To simplify loat{\sqrt{-92}}, we first factor out $-1$:

loat{\sqrt{-92}} = loat{\sqrt{-1 imes 92}} = loat{\sqrt{-1}} imes loat{\sqrt{92}} = iloat{\sqrt{92}}

Now, we need to simplify loat{\sqrt{92}}. We look for the largest perfect square that divides 92. The prime factorization of 92 is $2^2 imes 23$. Thus, we can rewrite loat{\sqrt{92}} as:

loat{\sqrt{92}} = loat{\sqrt{4 imes 23}} = loat{\sqrt{4}} imes loat{\sqrt{23}} = 2loat{\sqrt{23}}

Substituting this back into our expression, we get:

loat{\sqrt{-92}} = i imes 2loat{\sqrt{23}} = 2iloat{\sqrt{23}}

Now, our equation looks like this:

x = rac{12 loat{\pm} 2iloat{\sqrt{23}}}{2}

Simplifying the square root of a negative number is a crucial step in solving quadratic equations that have complex roots. By factoring out the imaginary unit and simplifying the remaining square root, we can express the solutions in their simplest form. This process showcases the importance of understanding complex numbers and their properties in algebra.

Final Simplification and Solutions

We have now arrived at the equation:

x = rac{12 loat{\pm} 2iloat{\sqrt{23}}}{2}

The final step is to simplify this expression by dividing both terms in the numerator by the denominator, which is 2. This gives us:

x = rac{12}{2} loat{\pm} rac{2iloat{\sqrt{23}}}{2}

Dividing 12 by 2, we get 6. Dividing 2iloat{\sqrt{23}} by 2, we get iloat{\sqrt{23}}. Therefore, the simplified solutions are:

x = 6 loat{\pm} iloat{\sqrt{23}}

These are the two complex solutions to the quadratic equation $x^2 - 12x + 59 = 0$. The solutions are complex because the discriminant ($b^2 - 4ac$) was negative, indicating that the parabola represented by the quadratic equation does not intersect the x-axis. The two solutions are a conjugate pair, which is a common characteristic of quadratic equations with complex roots. The solutions consist of a real part (6) and an imaginary part (loat{\pm} iloat{\sqrt{23}}). Understanding how to simplify complex numbers and identify complex roots is crucial in advanced algebra and calculus. The final simplified solutions x = 6 loat{\pm} iloat{\sqrt{23}} represent the values of $x$ that satisfy the original equation. This step-by-step solution demonstrates the application of the quadratic formula and the simplification of complex numbers, providing a comprehensive understanding of the solution process.

Conclusion

In conclusion, we have successfully solved the quadratic equation $x^2 - 12x + 59 = 0$ using the quadratic formula. The steps involved identifying the coefficients $a = 1$, $b = -12$, and $c = 59$, substituting these values into the formula, simplifying the resulting expression, and handling the complex numbers that arose due to the negative discriminant. The final solutions are x = 6 loat{\pm} iloat{\sqrt{23}}, which are complex conjugates. This process highlights the importance of the quadratic formula as a versatile tool for solving any quadratic equation, regardless of whether the roots are real or complex. Understanding how to work with complex numbers is a crucial skill in algebra and higher-level mathematics. By mastering the quadratic formula and the techniques for simplifying complex expressions, you can confidently tackle a wide range of algebraic problems. This comprehensive solution not only provides the answer but also reinforces the underlying concepts and methods used in solving quadratic equations. The ability to solve such equations is fundamental to many areas of mathematics, science, and engineering, making it an essential skill for students and professionals alike.