Solving $x^2 + 9x + 9 = 0$ A Comprehensive Guide

When faced with a quadratic equation in the form of $ax^2 + bx + c = 0$, our primary goal is to find the values of $x$ that satisfy the equation. These values are also known as the roots or solutions of the quadratic equation. There are several methods to solve quadratic equations, and in this comprehensive guide, we will explore the most commonly used techniques, including factoring, completing the square, and the quadratic formula. By understanding these methods, you will gain the ability to tackle a wide range of quadratic equations and confidently determine their solutions. Each method offers its unique approach, making it essential to grasp the underlying principles to choose the most suitable technique for a given equation. Let's delve into these methods step by step to equip you with the knowledge and skills necessary to solve quadratic equations effectively.

Methods for Solving Quadratic Equations

Factoring

Factoring is a powerful method for solving quadratic equations, but it's most effective when the equation can be easily factored. The basic idea behind factoring is to express the quadratic equation as a product of two binomials. When the product of two binomials equals zero, at least one of the binomials must be zero. By setting each binomial equal to zero, we can solve for the values of $x$ that satisfy the equation. To illustrate, consider the quadratic equation $x^2 + 5x + 6 = 0$. We can factor this equation into $(x + 2)(x + 3) = 0$. Now, we set each factor equal to zero: $x + 2 = 0$ and $x + 3 = 0$. Solving these linear equations, we find $x = -2$ and $x = -3$. These are the roots of the quadratic equation.

However, factoring isn't always straightforward. For more complex equations, it may be challenging to find the correct factors. In such cases, other methods like completing the square or the quadratic formula are more suitable. Factoring is best applied when the coefficients of the quadratic equation are integers and the roots are rational numbers. Understanding the patterns and techniques of factoring can significantly simplify the process of solving quadratic equations when it is applicable. Practice with various examples will enhance your ability to recognize factorable quadratic equations and apply this method effectively.

Completing the Square

Completing the square is a versatile method for solving quadratic equations, even when factoring is not straightforward. This technique involves manipulating the quadratic equation to create a perfect square trinomial on one side. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. By completing the square, we transform the equation into a form that allows us to easily solve for $x$. To illustrate the process, let's consider the quadratic equation $x^2 + 6x + 5 = 0$. First, we move the constant term to the right side of the equation: $x^2 + 6x = -5$. Next, we take half of the coefficient of the $x$ term (which is 6), square it (which gives us 9), and add it to both sides of the equation: $x^2 + 6x + 9 = -5 + 9$. This simplifies to $(x + 3)^2 = 4$. Now, we take the square root of both sides: $x + 3 = \\pm 2$. Solving for $x$, we get $x = -3 \\pm 2$, which gives us two solutions: $x = -1$ and $x = -5$.

Completing the square is particularly useful when the quadratic equation cannot be easily factored, or when the quadratic formula is not preferred. It provides a systematic approach to solving quadratic equations and is a fundamental technique in algebra. Mastering this method enhances your understanding of quadratic equations and provides you with another powerful tool to find solutions. While it may seem more involved than factoring in some cases, completing the square always works and is a reliable method for solving any quadratic equation.

The Quadratic Formula

The quadratic formula is a powerful and universally applicable method for solving quadratic equations of the form $ax^2 + bx + c = 0$. Unlike factoring and completing the square, the quadratic formula can be used to solve any quadratic equation, regardless of whether it has rational, irrational, or complex roots. This formula is derived from the process of completing the square and provides a direct way to find the roots of the equation. The quadratic formula is given by:

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

In this formula, $a$, $b$, and $c$ are the coefficients of the quadratic equation. The expression inside the square root, $b^2 - 4ac$, is known as the discriminant. The discriminant plays a crucial role in determining the nature of the roots. 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. To illustrate the use of the quadratic formula, let's consider the equation $2x^2 + 5x + 3 = 0$. Here, $a = 2$, $b = 5$, and $c = 3$. Plugging these values into the formula, we get:

$$x = \frac{-5 \pm \sqrt{5^2 - 4(2)(3)}}{2(2)} = \frac{-5 \pm \sqrt{25 - 24}}{4} = \frac{-5 \pm \sqrt{1}}{4} = \frac{-5 \pm 1}{4}$$

This gives us two solutions: $x = \frac{-5 + 1}{4} = -1$ and $x = \frac{-5 - 1}{4} = -\frac{3}{2}$.

The quadratic formula is an indispensable tool for solving quadratic equations. Its ability to handle any quadratic equation makes it a reliable and efficient method. Understanding and memorizing the quadratic formula is essential for anyone studying algebra and beyond. By mastering this formula, you can quickly and accurately find the solutions to a wide variety of quadratic equations, regardless of their complexity.

Solving the Given Equation: $x^2 + 9x + 9 = 0$

Now, let's apply these methods to solve the specific quadratic equation given: $x^2 + 9x + 9 = 0$. This equation is in the standard form $ax^2 + bx + c = 0$, where $a = 1$, $b = 9$, and $c = 9$. We will use the quadratic formula to find the solutions for $x$. The quadratic formula, as we discussed, is:

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

Plugging in the values $a = 1$, $b = 9$, and $c = 9$ into the formula, we get:

$$x = \frac{-9 \pm \sqrt{9^2 - 4(1)(9)}}{2(1)}$$

Now, we simplify the expression step by step. First, we calculate the value inside the square root:

$$9^2 - 4(1)(9) = 81 - 36 = 45$$

So, the equation becomes:

$$x = \frac{-9 \pm \sqrt{45}}{2}$$

Next, we simplify the square root. Since $45 = 9 \\times 5$, we can write $\sqrt{45}$ as $\sqrt{9 \\times 5} = \sqrt{9} \\times \sqrt{5} = 3\sqrt{5}$. Thus, the equation further simplifies to:

$$x = \frac{-9 \pm 3\sqrt{5}}{2}$$

This gives us two solutions for $x$:

$$x = \frac{-9 + 3\sqrt{5}}{2} \quad \text{and} \quad x = \frac{-9 - 3\sqrt{5}}{2}$$

Therefore, the solutions to the quadratic equation $x^2 + 9x + 9 = 0$ are $\frac{-9 + 3\sqrt{5}}{2}$ and $\frac{-9 - 3\sqrt{5}}{2}$. These are the exact values of $x$ that satisfy the equation. By using the quadratic formula, we have efficiently found the roots of the equation without needing to rely on factoring or completing the square.

Comparing with the Given Options

Comparing our solution $\frac{-9 \\pm 3\sqrt{5}}{2}$ with the given options, we can see that it matches option A:

A. $\frac{-9 \\pm 3\sqrt{5}}{2}$

So, the correct answer is option A.

Conclusion

In this comprehensive guide, we have explored various methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula. We have also applied the quadratic formula to solve the specific equation $x^2 + 9x + 9 = 0$ and found the solutions to be $\frac{-9 \\pm 3\sqrt{5}}{2}$. This solution matches option A from the given choices. Understanding these methods is crucial for mastering algebra and solving a wide range of mathematical problems. The quadratic formula, in particular, is a powerful tool that can be used to solve any quadratic equation, making it an essential technique for students and professionals alike. By practicing and applying these methods, you can enhance your problem-solving skills and confidently tackle quadratic equations.