Solve For $x$:${ \begin{array}{l} x^2 + 6x + 9 = 0 \ x = \square \end{array} }$

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 quadratic equations of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the quadratic formula to find the value of $x$ in the given equation.

What is a Quadratic Equation?

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

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

where $a$, $b$, and $c$ are constants, and $a \neq 0$. The value of $x$ that satisfies the equation is called the solution or root of the equation.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation. The quadratic formula is a general solution to the equation, and it works for all values of $a$, $b$, and $c$.

Solving the Given Equation

Let's apply the quadratic formula to the given equation:

$$x^2 + 6x + 9 = 0$$

In this equation, $a = 1$, $b = 6$, and $c = 9$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-6 \pm \sqrt{6^2 - 4(1)(9)}}{2(1)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-6 \pm \sqrt{36 - 36}}{2}$$

This simplifies to:

$$x = \frac{-6 \pm \sqrt{0}}{2}$$

Since the square root of zero is zero, we have:

$$x = \frac{-6 \pm 0}{2}$$

This simplifies to:

$$x = \frac{-6}{2}$$

Therefore, the solution to the equation is:

$$x = -3$$

Interpretation of the Solution

The solution $x = -3$ means that the value of $x$ that satisfies the equation is $-3$. This is the only solution to the equation, and it is a real number.

Conclusion

In this article, we have solved a quadratic equation using the quadratic formula. We have shown that the solution to the equation is $x = -3$. The quadratic formula is a powerful tool for solving quadratic equations, and it works for all values of $a$, $b$, and $c$. We hope that this article has provided a clear and concise explanation of how to solve quadratic equations.

Common Mistakes to Avoid

When solving quadratic equations, there are several common mistakes to avoid. These include:

• Not checking the discriminant: The discriminant is the expression under the square root in the quadratic formula. If the discriminant is negative, the equation has no real solutions.
• Not simplifying the expression: The quadratic formula involves simplifying the expression under the square root. Make sure to simplify the expression carefully to avoid errors.
• Not checking the solutions: After finding the solutions, make sure to check them to ensure that they satisfy the equation.

Real-World Applications

Quadratic equations have many real-world applications. Some examples include:

• Projectile motion: Quadratic equations are used to model the trajectory of projectiles, such as balls or rockets.
• Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Electrical engineering: Quadratic equations are used to analyze electrical circuits and find the values of resistors and capacitors.

Conclusion

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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 answer some of the most frequently asked questions about quadratic equations.

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 $x$ is two. The general form of a quadratic equation is:

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

where $a$, $b$, and $c$ are constants, and $a \neq 0$.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve the equation by setting each factor equal to zero.
• Quadratic formula: The quadratic formula is a general solution to the equation, and it is given by:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula. It is given by:

$$b^2 - 4ac$$

If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the number of solutions to a quadratic equation?

A: The number of solutions to a quadratic equation depends on the discriminant. If the discriminant is:

• Positive: The equation has two distinct real solutions.
• Zero: The equation has one real solution.
• Negative: The equation has no real solutions.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, you can use the quadratic formula to solve a quadratic equation with complex solutions. The quadratic formula will give you two complex solutions, which can be written in the form:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: How do I graph a quadratic equation?

A: To graph a quadratic equation, you can use the following steps:

• Find the x-intercepts: Set the equation equal to zero and solve for $x$ to find the x-intercepts.
• Find the vertex: The vertex of the parabola is the point where the parabola changes direction. You can find the vertex by using the formula:

$$x = -\frac{b}{2a}$$

• Plot the parabola: Use the x-intercepts and the vertex to plot the parabola.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have many real-world applications, including:

• Projectile motion: Quadratic equations are used to model the trajectory of projectiles, such as balls or rockets.
• Optimization: Quadratic equations are used to optimize functions, such as finding the maximum or minimum value of a function.
• Electrical engineering: Quadratic equations are used to analyze electrical circuits and find the values of resistors and capacitors.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. We hope that this article has provided a clear and concise explanation of some of the most frequently asked questions about quadratic equations.