Solve For $x$.$x^2 + 4x + 4 = 0$If There Is More Than One Solution, Separate Them With Commas. If There Is No Solution, Select No Solution.

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Introduction

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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 solve the equation $x^2 + 4x + 4 = 0$ and provide a step-by-step guide on how to solve quadratic equations.

Understanding Quadratic Equations

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A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. The quadratic equation can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Quadratic Formula

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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.

Solving the Equation $x^2 + 4x + 4 = 0$

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Now, let's apply the quadratic formula to solve the equation $x^2 + 4x + 4 = 0$. We have $a = 1$, $b = 4$, and $c = 4$. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

$$x = \frac{-4 \pm \sqrt{16 - 16}}{2}$$

$$x = \frac{-4 \pm \sqrt{0}}{2}$$

$$x = \frac{-4}{2}$$

$$x = -2$$

Therefore, the solution to the equation $x^2 + 4x + 4 = 0$ is $x = -2$.

Interpreting the Results

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When solving quadratic equations, we may encounter different types of solutions. The solution can be a single value, two values, or no solution at all. In this case, we obtained a single solution, $x = -2$. However, if the discriminant $b^2 - 4ac$ is positive, we will obtain two solutions. If the discriminant is zero, we will obtain a single solution. If the discriminant is negative, we will obtain no solution.

Conclusion

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Solving quadratic equations is an essential skill in mathematics. The quadratic formula is a powerful tool for solving quadratic equations. By applying the quadratic formula, we can solve equations of the form $ax^2 + bx + c = 0$. In this article, we solved the equation $x^2 + 4x + 4 = 0$ and obtained a single solution, $x = -2$. We also discussed the different types of solutions that can occur when solving quadratic equations.

Tips and Tricks

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• Always check the discriminant $b^2 - 4ac$ to determine the type of solution.
• Use the quadratic formula to solve quadratic equations.
• Simplify the expression to obtain the solution.
• Check the solution by plugging it back into the original equation.

Common Quadratic Equations

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Here are some common quadratic equations:

• $x^2 + 5x + 6 = 0$
• $x^2 - 3x - 4 = 0$
• $x^2 + 2x - 15 = 0$

Solving Quadratic Equations with Complex Solutions

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When solving quadratic equations, we may encounter complex solutions. A complex solution is a solution that involves the imaginary unit $i$, where $i = \sqrt{-1}$. To solve quadratic equations with complex solutions, we can use the quadratic formula and simplify the expression.

Example

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Solve the equation $x^2 + 2x + 2 = 0$.

We have $a = 1$, $b = 2$, and $c = 2$. Plugging these values into the quadratic formula, we get:

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

Simplifying the expression, we get:

$$x = \frac{-2 \pm \sqrt{4 - 8}}{2}$$

$$x = \frac{-2 \pm \sqrt{-4}}{2}$$

$$x = \frac{-2 \pm 2i}{2}$$

$$x = -1 \pm i$$

Therefore, the solutions to the equation $x^2 + 2x + 2 = 0$ are $x = -1 + i$ and $x = -1 - i$.

Conclusion

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Solving quadratic equations with complex solutions requires the use of the quadratic formula and simplification of the expression. By applying the quadratic formula, we can solve equations of the form $ax^2 + bx + c = 0$ and obtain complex solutions.

Final Thoughts

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Solving quadratic equations is an essential skill in mathematics. The quadratic formula is a powerful tool for solving quadratic equations. By applying the quadratic formula, we can solve equations of the form $ax^2 + bx + c = 0$ and obtain real or complex solutions. We hope this article has provided a comprehensive guide on solving quadratic equations and has helped you to improve your problem-solving skills.

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Introduction

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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 frequently asked questions about quadratic equations and provide a comprehensive guide on how to solve them.

Q: What is a quadratic equation?

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A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

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There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations and 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?

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The discriminant is the expression $b^2 - 4ac$ in the quadratic formula. It determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I determine the nature of the solutions?

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To determine the nature of the solutions, you need to calculate the discriminant and check its value. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

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

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A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. The general form of a linear equation is $ax + b = 0$, where $a$ and $b$ are constants.

Q: Can I solve a quadratic equation by factoring?

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Yes, you can solve a quadratic equation by factoring. However, factoring is not always possible, and the quadratic formula is a more general method for solving quadratic equations.

Q: What is the quadratic formula?

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The quadratic formula is a powerful tool for solving quadratic equations and 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: How do I apply the quadratic formula?

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To apply the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula and simplify the expression.

Q: What are some common quadratic equations?

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Some common quadratic equations include:

• $x^2 + 5x + 6 = 0$
• $x^2 - 3x - 4 = 0$
• $x^2 + 2x - 15 = 0$

Q: Can I solve a quadratic equation with complex solutions?

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Yes, you can solve a quadratic equation with complex solutions. To do this, you need to use the quadratic formula and simplify the expression.

Q: What is the difference between a real solution and a complex solution?

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A real solution is a solution that is a real number, while a complex solution is a solution that involves the imaginary unit $i$, where $i = \sqrt{-1}$.

Conclusion

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Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we have answered some frequently asked questions about quadratic equations and provided a comprehensive guide on how to solve them. We hope this article has helped you to improve your problem-solving skills and has provided a better understanding of quadratic equations.

Final Thoughts

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Solving quadratic equations is an essential skill in mathematics. The quadratic formula is a powerful tool for solving quadratic equations, and by applying it, you can solve equations of the form $ax^2 + bx + c = 0$ and obtain real or complex solutions. We hope this article has provided a comprehensive guide on solving quadratic equations and has helped you to improve your problem-solving skills.

Additional Resources

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If you want to learn more about quadratic equations, we recommend checking out the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equations
• Wolfram Alpha: Quadratic Equations

Practice Problems

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Here are some practice problems to help you improve your skills in solving quadratic equations:

• Solve the equation $x^2 + 4x + 4 = 0$.
• Solve the equation $x^2 - 2x - 15 = 0$.
• Solve the equation $x^2 + 3x + 2 = 0$.

Conclusion

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Solving quadratic equations is an essential skill in mathematics. The quadratic formula is a powerful tool for solving quadratic equations, and by applying it, you can solve equations of the form $ax^2 + bx + c = 0$ and obtain real or complex solutions. We hope this article has provided a comprehensive guide on solving quadratic equations and has helped you to improve your problem-solving skills.