Solve The Quadratic Equation: $x^2 + 10x + 13 = 4$

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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 a specific quadratic equation: $x^2 + 10x + 13 = 4$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and mathematical techniques.

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 (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. In our specific equation, $x^2 + 10x + 13 = 4$, we can rewrite it as:

$$x^2 + 10x + 9 = 0$$

The Quadratic Formula

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The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

In our case, $a = 1$, $b = 10$, and $c = 9$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-10 \pm \sqrt{10^2 - 4(1)(9)}}{2(1)}$$

Simplifying the Quadratic Formula

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To simplify the quadratic formula, we need to evaluate the expression under the square root:

$$10^2 - 4(1)(9) = 100 - 36 = 64$$

So, the quadratic formula becomes:

$$x = \frac{-10 \pm \sqrt{64}}{2}$$

Solving for $x$

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Now, we can solve for $x$ by evaluating the expression:

$$x = \frac{-10 \pm \sqrt{64}}{2}$$

$$x = \frac{-10 \pm 8}{2}$$

We have two possible solutions:

$$x_1 = \frac{-10 + 8}{2} = -1$$

$$x_2 = \frac{-10 - 8}{2} = -9$$

Verifying the Solutions

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To verify our solutions, we can plug them back into the original equation:

$$x^2 + 10x + 9 = 0$$

For $x = -1$:

$$(-1)^2 + 10(-1) + 9 = 1 - 10 + 9 = 0$$

For $x = -9$:

$$(-9)^2 + 10(-9) + 9 = 81 - 90 + 9 = 0$$

Both solutions satisfy the original equation, so we can be confident in our answer.

Conclusion

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Solving quadratic equations can be a challenging task, but with the right techniques and tools, it can be done. In this article, we used the quadratic formula to solve the equation $x^2 + 10x + 13 = 4$. We broke down the solution into manageable steps, using algebraic manipulations and mathematical techniques. By following these steps, you can solve quadratic equations with confidence.

Tips and Tricks

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• Always check your solutions by plugging them back into the original equation.
• Use the quadratic formula to solve quadratic equations.
• Simplify the quadratic formula by evaluating the expression under the square root.
• Solve for $x$ by evaluating the expression.

Common Mistakes

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• Failing to check solutions by plugging them back into the original equation.
• Not using the quadratic formula to solve quadratic equations.
• Not simplifying the quadratic formula by evaluating the expression under the square root.
• Not solving for $x$ by evaluating the expression.

Real-World Applications

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Quadratic equations have many real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design bridges and other structures.
• Economics: Quadratic equations are used to model the behavior of economic systems.

Final Thoughts

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Solving quadratic equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with confidence. Remember to always check your solutions, use the quadratic formula, simplify the formula, and solve for $x$. With practice and patience, you can master the art of solving quadratic equations.

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Introduction

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Quadratic equations can be a challenging topic for many students and professionals. In this article, we will address some of the most frequently asked questions about quadratic equations, providing clear and concise answers to help you better understand this important mathematical concept.

Q: What is a quadratic equation?

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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:

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

Q: How do I solve a quadratic equation?

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A: There are several methods to solve quadratic equations, including:

• Factoring: If the quadratic expression can be factored into the product of two binomials, you can solve for $x$ by setting each factor equal to zero.
• Quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions are given by:

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

Q: What is the quadratic formula?

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A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

Q: How do I use the quadratic formula?

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A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. Then, simplify the expression under the square root and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

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A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic expression as the product of two binomials, while the quadratic formula involves using a mathematical formula to find the solutions.

Q: Can I use the quadratic formula to solve all quadratic equations?

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A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it may not always be the most efficient method, especially for equations that can be easily factored.

Q: How do I check my solutions?

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A: To check your solutions, plug them back into the original equation and verify that they satisfy the equation.

Q: What are some common mistakes to avoid when solving quadratic equations?

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A: Some common mistakes to avoid when solving quadratic equations include:

• Failing to check solutions by plugging them back into the original equation.
• Not using the quadratic formula to solve quadratic equations.
• Not simplifying the quadratic formula by evaluating the expression under the square root.
• Not solving for $x$ by evaluating the expression.

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

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

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
• Engineering: Quadratic equations are used to design bridges and other structures.
• Economics: Quadratic equations are used to model the behavior of economic systems.

Conclusion

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Quadratic equations can be a challenging topic, but with practice and patience, you can master the art of solving them. By following the steps outlined in this article, you can solve quadratic equations with confidence. Remember to always check your solutions, use the quadratic formula, simplify the formula, and solve for $x$. With this knowledge, you can tackle a wide range of mathematical problems and apply quadratic equations to real-world situations.

Additional Resources

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For further learning and practice, we recommend the following resources:

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

Final Thoughts

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Solving quadratic equations is a crucial skill for students and professionals alike. By following the steps outlined in this article and practicing with real-world examples, you can master the art of solving quadratic equations and apply them to a wide range of mathematical problems.