Solve For K K K . K 2 − 14 K = 0 K^2 - 14k = 0 K 2 − 14 K = 0 Write Each Solution As An Integer, Proper Fraction, Or Improper Fraction In Simplest Form. If There Are Multiple Solutions, Separate Them With Commas. K = □ K = \square K = □

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we will focus on solving the quadratic equation $k^2 - 14k = 0$ and present the solutions in their simplest form.

Understanding the Equation

The given equation is a quadratic equation in the form of $ax^2 + bx + c = 0$, where $a = 1$, $b = -14$, and $c = 0$. To solve this equation, we can use the factoring method, the quadratic formula, or the completing the square method.

Factoring Method

One way to solve the equation is by factoring. We can rewrite the equation as $k(k - 14) = 0$. This means that either $k = 0$ or $k - 14 = 0$. Solving for $k$ in both cases, we get $k = 0$ and $k = 14$.

Quadratic Formula

Another way to solve the equation is by using the quadratic formula. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. In this case, $a = 1$, $b = -14$, and $c = 0$. Plugging these values into the formula, we get $k = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(0)}}{2(1)}$. Simplifying the expression, we get $k = \frac{14 \pm \sqrt{196}}{2}$. This can be further simplified to $k = \frac{14 \pm 14}{2}$.

Completing the Square Method

The completing the square method involves rewriting the equation in a perfect square form. We can rewrite the equation as $k^2 - 14k + 49 = 49$. This can be further simplified to $(k - 7)^2 = 49$. Taking the square root of both sides, we get $k - 7 = \pm 7$. Solving for $k$, we get $k = 7 \pm 7$.

Solutions

Using the factoring method, we found that $k = 0$ and $k = 14$. Using the quadratic formula, we found that $k = \frac{14 \pm 14}{2}$. Using the completing the square method, we found that $k = 7 \pm 7$. Combining all the solutions, we get $k = 0, 14, 7 + 7, 7 - 7$. Simplifying the last two solutions, we get $k = 14, 0, 14, 0$.

Conclusion

In this article, we solved the quadratic equation $k^2 - 14k = 0$ using the factoring method, the quadratic formula, and the completing the square method. We presented the solutions in their simplest form and combined all the solutions to get the final answer.

Final Answer

The final answer is $k = 0, 14$.

Additional Tips and Tricks

• When solving quadratic equations, it is essential to check the solutions by plugging them back into the original equation.
• The quadratic formula can be used to solve quadratic equations that cannot be factored.
• The completing the square method can be used to rewrite quadratic equations in a perfect square form.

Real-World Applications

Quadratic equations have numerous 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 economic systems and make predictions about future trends.

Common Mistakes to Avoid

• When solving quadratic equations, it is essential to check the solutions by plugging them back into the original equation.
• The quadratic formula should be used carefully, as it can lead to complex solutions.
• The completing the square method should be used carefully, as it can lead to incorrect solutions.

Conclusion

In conclusion, solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to approach these types of problems. In this article, we solved the quadratic equation $k^2 - 14k = 0$ using the factoring method, the quadratic formula, and the completing the square method. We presented the solutions in their simplest form and combined all the solutions to get the final answer.

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Introduction

In our previous article, we solved the quadratic equation $k^2 - 14k = 0$ using the factoring method, the quadratic formula, and the completing the square method. In this article, we will answer some frequently asked questions about solving quadratic equations.

Q&A

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

A: A quadratic equation is a polynomial equation of degree two, which means that the highest power of the variable is two. A linear equation, on the other hand, is a polynomial equation of degree one, which means that the highest power of the variable is one.

Q: How do I know which method to use to solve a quadratic equation?

A: The choice of method depends on the specific equation and the type of solutions you are looking for. If the equation can be factored easily, the factoring method may be the best choice. If the equation cannot be factored easily, the quadratic formula or the completing the square method may be more suitable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that can be used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the completing the square method?

A: The completing the square method is a technique that can be used to rewrite a quadratic equation in a perfect square form. This can be useful for solving quadratic equations that cannot be factored easily.

Q: How do I check my solutions?

A: To check your solutions, plug them back into the original equation and simplify. If the equation is true, then the solution is correct. If the equation is not true, then the solution is incorrect.

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

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

• Not checking the solutions by plugging them back into the original equation
• Using the quadratic formula without checking for complex solutions
• Using the completing the square method without checking for incorrect solutions

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

A: Quadratic equations have numerous 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 economic systems and make predictions about future trends.

Q: How do I simplify complex solutions?

A: To simplify complex solutions, use the fact that $i = \sqrt{-1}$. For example, if you have a solution of the form $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, you can simplify it by using the fact that $i = \sqrt{-1}$.

Q: What is the difference between a proper fraction and an improper fraction?

A: A proper fraction is a fraction where the numerator is less than the denominator, such as $\frac{1}{2}$. An improper fraction is a fraction where the numerator is greater than or equal to the denominator, such as $\frac{3}{2}$.

Q: How do I convert a mixed number to an improper fraction?

A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Then, write the result as a fraction with the denominator as the original denominator. For example, to convert the mixed number $2\frac{1}{2}$ to an improper fraction, multiply the whole number by the denominator and add the numerator: $2 \times 2 + 1 = 5$. Then, write the result as a fraction with the denominator as the original denominator: $\frac{5}{2}$.

Conclusion

In this article, we answered some frequently asked questions about solving quadratic equations. We covered topics such as the difference between quadratic and linear equations, the choice of method for solving quadratic equations, and the real-world applications of quadratic equations. We also covered some common mistakes to avoid when solving quadratic equations and how to simplify complex solutions.