Solve For \[$ X \$\] In The Equation $ X^2 - 12x + 59 = 0 $.A. \[$ X = -12 \pm \sqrt{85} \$\]B. \[$ X = -6 \pm \sqrt{23}i \$\]C. \[$ X = 6 \pm \sqrt{23}i \$\]D. \[$ X = 12 \pm \sqrt{85} \$\]

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $x^2 - 12x + 59 = 0$, and explore the different methods and techniques used to find the solutions.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

The Quadratic Formula

One of the most common methods for solving quadratic equations is the quadratic formula. The quadratic formula is given by:

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

This formula provides two solutions for the equation, which are given by the plus and minus signs.

Applying the Quadratic Formula

Now, let's apply the quadratic formula to the given equation, $x^2 - 12x + 59 = 0$. We have:

$$a = 1$$

$$b = -12$$

$$c = 59$$

Substituting these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

$$x = \frac{12 \pm \sqrt{144 - 236}}{2}$$

$$x = \frac{12 \pm \sqrt{-92}}{2}$$

$$x = \frac{12 \pm 2\sqrt{-23}}{2}$$

$$x = 6 \pm \sqrt{-23}$$

Complex Solutions

Notice that the solution involves the square root of a negative number, which means the solutions are complex numbers. In this case, the solutions are given by:

$$x = 6 \pm \sqrt{-23}$$

This can be rewritten as:

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

where $i$ is the imaginary unit, which satisfies $i^2 = -1$.

Conclusion

In this article, we solved the quadratic equation $x^2 - 12x + 59 = 0$ using the quadratic formula. We found that the solutions are complex numbers, given by $x = 6 \pm i\sqrt{23}$. This demonstrates the importance of understanding complex numbers and their applications in mathematics.

Comparison of Options

Now, let's compare our solution with the given options:

A. $x = -12 \pm \sqrt{85}$ B. $x = -6 \pm \sqrt{23}i$ C. $x = 6 \pm \sqrt{23}i$ D. $x = 12 \pm \sqrt{85}$

Our solution matches option C, which is:

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

Therefore, the correct answer is:

Introduction

In our previous article, we solved the quadratic equation $x^2 - 12x + 59 = 0$ using the quadratic formula. We found that the solutions are complex numbers, given by $x = 6 \pm i\sqrt{23}$. In this article, we will answer some frequently asked questions about quadratic equations and provide additional insights and examples.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, we can set each binomial equal to zero and solve for the variable.
• Quadratic formula: The quadratic formula is given by:

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

This formula provides two solutions for the equation, which are given by the plus and minus signs.

• Graphing: We can graph the quadratic function and find the x-intercepts, which correspond to the solutions of the equation.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that provides two solutions for a quadratic equation. It is given by:

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

This formula is a powerful tool for solving quadratic equations, and it is widely used in mathematics and science.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a highest power of two, while a linear equation has a highest power of one.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions. This is because the quadratic formula provides two solutions, and there are no other possible solutions.

Q: Can a quadratic equation have no solutions?

A: Yes, a quadratic equation can have no solutions. This occurs when the discriminant ($b^2 - 4ac$) is negative, which means that the quadratic expression has no real roots.

Q: What is the discriminant?

A: The discriminant is the expression $b^2 - 4ac$ that appears in the quadratic formula. It determines the nature of the solutions of the quadratic equation.

Q: Can the discriminant be zero?

A: Yes, the discriminant can be zero. This occurs when the quadratic equation has a repeated root, which means that the two solutions are the same.

Q: What is the significance of the discriminant?

A: The discriminant determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the solutions are real and distinct. If the discriminant is zero, the solutions are real and repeated. If the discriminant is negative, the solutions are complex.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations and provided additional insights and examples. We hope that this article has been helpful in clarifying the concepts and techniques involved in solving quadratic equations.