Solve The Equation For X X X . 5 X 2 − 4 X = 6 5x^2 - 4x = 6 5 X 2 − 4 X = 6 A. X = − 2 ± 26 5 X = \frac{-2 \pm \sqrt{26}}{5} X = 5 − 2 ± 26 ​ ​ B. X = − 2 ± 34 5 X = \frac{-2 \pm \sqrt{34}}{5} X = 5 − 2 ± 34 ​ ​ C. X = 2 ± 26 5 X = \frac{2 \pm \sqrt{26}}{5} X = 5 2 ± 26 ​ ​ D. X = 2 ± 34 5 X = \frac{2 \pm \sqrt{34}}{5} X = 5 2 ± 34 ​ ​

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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 delve into the world of quadratic equations and provide a step-by-step guide on how to solve them. We will focus on the equation $5x^2 - 4x = 6$ and explore the different methods of solving it.

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. Quadratic equations can be solved using various methods, including factoring, the quadratic formula, and graphing.

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. To use the quadratic formula, we need to identify the values of $a$, $b$, and $c$ in the given equation.

Solving the Equation $5x^2 - 4x = 6$

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To solve the equation $5x^2 - 4x = 6$, we need to first rewrite it in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting $6$ from both sides of the equation:

$$5x^2 - 4x - 6 = 0$$

Now we have a quadratic equation in the standard form. We can identify the values of $a$, $b$, and $c$ as follows:

• $a = 5$
• $b = -4$
• $c = -6$

Applying the Quadratic Formula

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Now that we have identified the values of $a$, $b$, and $c$, we can apply the quadratic formula to solve the equation. Plugging in the values, we get:

$$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(5)(-6)}}{2(5)}$$

Simplifying the expression, we get:

$$x = \frac{4 \pm \sqrt{16 + 120}}{10}$$

$$x = \frac{4 \pm \sqrt{136}}{10}$$

$$x = \frac{4 \pm 2\sqrt{34}}{10}$$

$$x = \frac{2 \pm \sqrt{34}}{5}$$

Conclusion

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In this article, we have solved the quadratic equation $5x^2 - 4x = 6$ using the quadratic formula. We have identified the values of $a$, $b$, and $c$ and applied the quadratic formula to solve the equation. The solution to the equation is $x = \frac{2 \pm \sqrt{34}}{5}$.

Discussion

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The quadratic formula is a powerful tool for solving quadratic equations. It is a general method that can be applied to any quadratic equation, regardless of whether it can be factored or not. The quadratic formula is a fundamental concept in mathematics and is used extensively in various fields, including physics, engineering, and economics.

Final Answer

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The final answer to the equation $5x^2 - 4x = 6$ is:

• $x = \frac{2 \pm \sqrt{34}}{5}$

This is the correct solution to the equation, and it is the only solution that satisfies the equation.

References

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• [1] "Quadratic Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/quadratic.html
• [2] "Quadratic Formula" by Khan Academy. Retrieved from https://www.khanacademy.org/math/algebra/x2f-quadratic-formula/x2f-quadratic-formula/v/quadratic-formula

Note: The references provided are for informational purposes only and are not intended to be a comprehensive list of resources on the topic.

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Introduction

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Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task 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 topic.

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.

Q: How do I solve a quadratic equation?

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A: There are several methods to solve a quadratic equation, including factoring, the quadratic formula, and graphing. The quadratic formula is a powerful tool that can be used to solve any quadratic equation, regardless of whether it can be factored or not.

Q: What is the quadratic formula?

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A: The quadratic formula is a formula that can be used to solve a quadratic equation. 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: How do I apply the quadratic formula?

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A: To apply the quadratic formula, you need to identify the values of $a$, $b$, and $c$ in the given equation. Then, plug these values into the quadratic formula and simplify the expression to find the solutions.

Q: What are the solutions to a quadratic equation?

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A: The solutions to a quadratic equation are the values of $x$ that satisfy the equation. These solutions can be real or complex numbers.

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

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A: The number of solutions to a quadratic equation depends on the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

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A: The discriminant is the expression under the square root in the quadratic formula, given by $b^2 - 4ac$. It determines the number of solutions to a quadratic equation.

Q: How do I graph a quadratic equation?

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A: To graph a quadratic equation, you can use a graphing calculator or a computer program. You can also use a table of values to plot the graph.

Q: What are the applications of quadratic equations?

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A: Quadratic equations have numerous applications in various fields, including physics, engineering, economics, and computer science. They are used to model real-world phenomena, such as the motion of objects, the growth of populations, and the behavior of electrical circuits.

Q: Can I use quadratic equations to solve real-world problems?

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A: Yes, quadratic equations can be used to solve real-world problems. They are a powerful tool for modeling and analyzing complex systems.

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:

• Not identifying the values of $a$, $b$, and $c$ correctly
• Not applying the quadratic formula correctly
• Not simplifying the expression correctly
• Not checking the solutions for validity

Q: How can I practice solving quadratic equations?

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A: You can practice solving quadratic equations by working through examples and exercises in a textbook or online resource. You can also use a graphing calculator or a computer program to visualize the solutions.

Q: What are some resources for learning more about quadratic equations?

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A: Some resources for learning more about quadratic equations include:

• Textbooks on algebra and mathematics
• Online resources, such as Khan Academy and Math Open Reference
• Graphing calculators and computer programs
• Online communities and forums for mathematics enthusiasts

Conclusion

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Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. By understanding the quadratic formula and its applications, you can solve quadratic equations with confidence and accuracy. Remember to practice regularly and seek help when needed to become proficient in solving quadratic equations.