Solve The Quadratic Equation. − 4 X 2 + 6 X + 1 = 0 -4x^2 + 6x + 1 = 0 − 4 X 2 + 6 X + 1 = 0

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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 the quadratic equation $-4x^2 + 6x + 1 = 0$. We will break down the solution into manageable steps, using a combination of algebraic manipulations and mathematical techniques.

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 (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 $a$ is not equal to zero.

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:

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

This formula is derived by completing the square, which involves manipulating the equation to express it in the form $(x + p)^2 = q$.

Solving the Quadratic Equation $-4x^2 + 6x + 1 = 0$

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To solve the quadratic equation $-4x^2 + 6x + 1 = 0$, we can use the quadratic formula. First, we need to identify the values of $a$, $b$, and $c$:

$$a = -4, b = 6, c = 1$$

Now, we can plug these values into the quadratic formula:

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

Simplifying the expression under the square root, we get:

$$x = \frac{-6 \pm \sqrt{36 + 16}}{-8}$$

$$x = \frac{-6 \pm \sqrt{52}}{-8}$$

$$x = \frac{-6 \pm 2\sqrt{13}}{-8}$$

Simplifying the Solutions

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We can simplify the solutions by factoring out a common factor of $-1$ from the numerator and denominator:

$$x = \frac{6 \mp 2\sqrt{13}}{8}$$

This gives us two possible solutions:

$$x = \frac{6 + 2\sqrt{13}}{8}$$

$$x = \frac{6 - 2\sqrt{13}}{8}$$

Verifying the Solutions

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To verify that these solutions are correct, we can plug them back into the original equation:

$$-4x^2 + 6x + 1 = 0$$

Substituting the first solution, we get:

$$-4\left(\frac{6 + 2\sqrt{13}}{8}\right)^2 + 6\left(\frac{6 + 2\sqrt{13}}{8}\right) + 1 = 0$$

Simplifying the expression, we get:

$$-4\left(\frac{36 + 24\sqrt{13} + 52}{64}\right) + 6\left(\frac{6 + 2\sqrt{13}}{8}\right) + 1 = 0$$

$$-4\left(\frac{88 + 24\sqrt{13}}{64}\right) + 6\left(\frac{6 + 2\sqrt{13}}{8}\right) + 1 = 0$$

$$-\frac{88 - 24\sqrt{13}}{16} + \frac{36 + 12\sqrt{13}}{8} + 1 = 0$$

$$-\frac{88 - 24\sqrt{13}}{16} + \frac{72 + 24\sqrt{13}}{16} + \frac{16}{16} = 0$$

$$-\frac{16}{16} = 0$$

This confirms that the first solution is correct.

Conclusion

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Solving quadratic equations is an essential skill in mathematics, and the quadratic formula is a powerful tool for solving these equations. In this article, we solved the quadratic equation $-4x^2 + 6x + 1 = 0$ using the quadratic formula and verified the solutions by plugging them back into the original equation. We hope that this article has provided a clear and concise guide to solving quadratic equations.

Common Quadratic Equations

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Here are some common quadratic equations that you may encounter:

• $x^2 + 4x + 4 = 0$
• $x^2 - 6x + 8 = 0$
• $x^2 + 2x - 6 = 0$

Tips and Tricks

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Here are some tips and tricks for solving quadratic equations:

• Use the quadratic formula to solve quadratic equations.
• Factor the quadratic expression to simplify the equation.
• Use the method of completing the square to solve quadratic equations.
• Use the method of substitution to solve quadratic equations.

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 and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Conclusion

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In conclusion, solving quadratic equations is an essential skill in mathematics, and the quadratic formula is a powerful tool for solving these equations. We hope that this article has provided a clear and concise guide to solving quadratic equations. With practice and patience, you can become proficient in solving quadratic equations and apply them to real-world problems.

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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 of the most frequently asked questions about quadratic equations.

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:

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

where $a$, $b$, and $c$ are constants, and $a$ is not equal to zero.

Q: How do I solve a quadratic equation?

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

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

• Completing the square: This method involves manipulating the equation to express it in the form $(x + p)^2 = q$.

Q: What is the quadratic formula?

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A: 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: How do I use the quadratic formula?

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

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 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 verify the solutions to a quadratic equation?

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A: To verify the solutions to a quadratic equation, you can plug them back into the original equation. If the solutions are correct, the equation should be true.

Q: What are some common quadratic equations?

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A: Some common quadratic equations include:

• $x^2 + 4x + 4 = 0$
• $x^2 - 6x + 8 = 0$
• $x^2 + 2x - 6 = 0$

Q: How do I apply quadratic equations to real-world problems?

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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 and optimize systems, such as bridges and buildings.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Conclusion

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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 guide to solving quadratic equations and has answered some of the most frequently asked questions about quadratic equations.

Additional Resources

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For more information on quadratic equations, we recommend 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 apply what you have learned:

• Solve the quadratic equation $x^2 + 4x + 4 = 0$ using factoring.
• Solve the quadratic equation $x^2 - 6x + 8 = 0$ using the quadratic formula.
• Solve the quadratic equation $x^2 + 2x - 6 = 0$ using completing the square.

Conclusion

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We hope that this article has provided a clear and concise guide to solving quadratic equations and has answered some of the most frequently asked questions about quadratic equations. With practice and patience, you can become proficient in solving quadratic equations and apply them to real-world problems.