Solve The Quadratic Equation:$\[ 12y^2 - 17y - 5 = 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 $12y^2 - 17y - 5 = 0$. We will break down the solution into manageable steps, using a combination of algebraic techniques and mathematical concepts.

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, $y$) is two. The general form of a quadratic equation is:

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

where $a$, $b$, and $c$ are constants, and $x$ is the variable. In our example, the quadratic equation is:

$$12y^2 - 17y - 5 = 0$$

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 example, we have $a = 12$, $b = -17$, and $c = -5$. Plugging these values into the quadratic formula, we get:

$$y = \frac{-(-17) \pm \sqrt{(-17)^2 - 4(12)(-5)}}{2(12)}$$

Simplifying the Quadratic Formula

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To simplify the quadratic formula, we can start by evaluating the expression inside the square root:

$$(-17)^2 - 4(12)(-5) = 289 + 240 = 529$$

Now, we can rewrite the quadratic formula as:

$$y = \frac{17 \pm \sqrt{529}}{24}$$

Simplifying the Square Root

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The square root of 529 is 23, so we can rewrite the quadratic formula as:

$$y = \frac{17 \pm 23}{24}$$

Solving for $y$

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Now, we can solve for $y$ by evaluating the two possible values:

$$y_1 = \frac{17 + 23}{24} = \frac{40}{24} = \frac{5}{3}$$

$$y_2 = \frac{17 - 23}{24} = \frac{-6}{24} = -\frac{1}{4}$$

Conclusion

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In this article, we solved the quadratic equation $12y^2 - 17y - 5 = 0$ using the quadratic formula. We broke down the solution into manageable steps, using a combination of algebraic techniques and mathematical concepts. We simplified the quadratic formula, evaluated the expression inside the square root, and solved for $y$ by evaluating the two possible values. The solutions to the quadratic equation are $y = \frac{5}{3}$ and $y = -\frac{1}{4}$.

Applications of Quadratic Equations

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Quadratic equations have numerous applications in mathematics, science, and engineering. Some examples include:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Computer Science: Quadratic equations are used in algorithms for solving systems of linear equations, finding the shortest path in a graph, and other applications.

Tips and Tricks for Solving Quadratic Equations

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Here are some tips and tricks for solving quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the solutions to any quadratic equation.
• Simplify the quadratic formula: Before solving for $y$, simplify the quadratic formula by evaluating the expression inside the square root.
• Check for extraneous solutions: When solving for $y$, check for extraneous solutions by plugging the solutions back into the original equation.
• Use algebraic techniques: Algebraic techniques, such as factoring and completing the square, can be used to solve quadratic equations.

Common Mistakes to Avoid

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Here are some common mistakes to avoid when solving quadratic equations:

• Not simplifying the quadratic formula: Failing to simplify the quadratic formula can lead to incorrect solutions.
• Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.
• Not using algebraic techniques: Failing to use algebraic techniques can make solving quadratic equations more difficult.

Conclusion

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In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By using the quadratic formula, simplifying the quadratic formula, and checking for extraneous solutions, we can solve quadratic equations with ease. Remember to use algebraic techniques, such as factoring and completing the square, to make solving quadratic equations more efficient. 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, $y$) 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 for solving quadratic equations, including:

• Factoring: If the quadratic equation can be factored, we can set each factor equal to zero and solve for the variable.
• 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?

---

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 simplify the quadratic formula?

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A: To simplify the quadratic formula, we can start by evaluating the expression inside the square root:

$$b^2 - 4ac$$

If this expression is positive, we can take the square root and simplify the formula. If the expression is negative, we will have complex solutions.

Q: What are complex solutions?

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A: Complex solutions are solutions that involve the imaginary unit, $i$. They are of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit.

Q: How do I check for extraneous solutions?

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A: To check for extraneous solutions, we can plug the solutions back into the original equation. If the solution is not a valid solution, it is an extraneous solution.

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 simplifying the quadratic formula: Failing to simplify the quadratic formula can lead to incorrect solutions.
• Not checking for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.
• Not using algebraic techniques: Failing to use algebraic techniques can make solving quadratic equations more difficult.

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

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A: Quadratic equations have numerous applications in mathematics, science, and engineering. Some examples include:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Computer Science: Quadratic equations are used in algorithms for solving systems of linear equations, finding the shortest path in a graph, and other applications.

Q: What are some tips and tricks for solving quadratic equations?

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A: Some tips and tricks for solving quadratic equations include:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
• Simplify the quadratic formula: Before solving for $y$, simplify the quadratic formula by evaluating the expression inside the square root.
• Check for extraneous solutions: When solving for $y$, check for extraneous solutions by plugging the solutions back into the original equation.
• Use algebraic techniques: Algebraic techniques, such as factoring and completing the square, can be used to solve quadratic equations.

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. By using the quadratic formula, simplifying the quadratic formula, and checking for extraneous solutions, we can solve quadratic equations with ease. Remember to use algebraic techniques, such as factoring and completing the square, to make solving quadratic equations more efficient. With practice and patience, you can become proficient in solving quadratic equations and apply them to real-world problems.