Solve For { W $}$ In The Equation:${ 6w^2 + W - 15 = 0 }$ { W = \square \}

Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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 this article, we will focus on solving quadratic equations of the form $ax^2 + bx + c = 0$, with a specific emphasis on the equation $6w^2 + w - 15 = 0$.

Understanding the Equation $6w^2 + w - 15 = 0$

The given equation is $6w^2 + w - 15 = 0$. To solve for $w$, we need to find the values of $w$ that satisfy this equation. The equation is a quadratic equation, and it can be solved using various methods such as factoring, completing the square, or using the quadratic formula.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

In this case, $a = 6$, $b = 1$, and $c = -15$. Plugging these values into the quadratic formula, we get:

$$w = \frac{-1 \pm \sqrt{1^2 - 4(6)(-15)}}{2(6)}$$

Simplifying the Expression

To simplify the expression, we need to evaluate the expression inside the square root:

$$1^2 - 4(6)(-15) = 1 + 360 = 361$$

Now, we can rewrite the expression as:

$$w = \frac{-1 \pm \sqrt{361}}{12}$$

Evaluating the Square Root

The square root of 361 is 19, so we can rewrite the expression as:

$$w = \frac{-1 \pm 19}{12}$$

Solving for w

Now, we have two possible solutions for $w$:

$$w = \frac{-1 + 19}{12} = \frac{18}{12} = \frac{3}{2}$$

$$w = \frac{-1 - 19}{12} = \frac{-20}{12} = -\frac{5}{3}$$

Conclusion

In this article, we solved the quadratic equation $6w^2 + w - 15 = 0$ using the quadratic formula. We found two possible solutions for $w$, which are $w = \frac{3}{2}$ and $w = -\frac{5}{3}$. These solutions satisfy the given equation, and they can be used to find the values of $w$ that make the equation true.

Importance of Quadratic Equations

Quadratic equations are an essential part of mathematics, and they have numerous applications in various fields. They are used to model real-world problems, such as the motion of objects, the growth of populations, and the behavior of electrical circuits. In addition, quadratic equations are used in computer science, engineering, and economics to solve problems and make predictions.

Real-World Applications of Quadratic Equations

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects, such as the trajectory of a projectile or the motion of a pendulum.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electrical circuits.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as the supply and demand of goods and services.
• Computer Science: Quadratic equations are used to solve problems and make predictions in computer science, such as the behavior of algorithms and the performance of computer systems.

Tips for Solving Quadratic Equations

Solving quadratic equations can be challenging, but there are several tips that can help:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the values of the variable that satisfy the equation.
• Factor the equation: Factoring the equation can make it easier to solve. Look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the variable.
• Use completing the square: Completing the square is a method for solving quadratic equations. It involves rewriting the equation in a form that allows us to easily find the values of the variable.
• Use graphing: Graphing the equation can help us visualize the solutions and make it easier to find the values of the variable.

Conclusion

In conclusion, solving quadratic equations is an essential part of mathematics, and it has numerous applications in various fields. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to find the values of the variable that satisfy the equation. By following the tips and techniques outlined in this article, we can solve quadratic equations and make predictions and solve problems in various fields.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In this article, we will answer some frequently asked questions about quadratic equations, including how to solve them, what they are used for, and how to apply them in real-world situations.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable 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.

Q: How do I solve a quadratic equation?

A: There are several methods for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. The quadratic formula is a powerful tool for solving quadratic equations, and it is given by:

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

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations. It is given by:

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

Q: How do I use the quadratic formula?

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

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, such as the trajectory of a projectile or the motion of a pendulum.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electrical circuits.
• Economics: Quadratic equations are used to model the behavior of economic systems, such as the supply and demand of goods and services.
• Computer Science: Quadratic equations are used to solve problems and make predictions in computer science, such as the behavior of algorithms and the performance of computer systems.

Q: How do I factor a quadratic equation?

A: Factoring a quadratic equation involves finding two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the variable. For example, if we have the quadratic equation $x^2 + 5x + 6 = 0$, we can factor it as $(x + 3)(x + 2) = 0$.

Q: What is completing the square?

A: Completing the square is a method for solving quadratic equations. It involves rewriting the equation in a form that allows us to easily find the values of the variable.

Q: How do I use completing the square?

A: To use completing the square, you need to rewrite the quadratic equation in the form $(x + \frac{b}{2})^2 = \frac{b^2 - 4ac}{4}$. Then, you need to simplify the expression and solve for $x$.

Q: What are some tips for solving quadratic equations?

A: Here are some tips for solving quadratic equations:

• Use the quadratic formula: The quadratic formula is a powerful tool for solving quadratic equations.
• Factor the equation: Factoring the equation can make it easier to solve.
• Use completing the square: Completing the square is a method for solving quadratic equations.
• Use graphing: Graphing the equation can help us visualize the solutions and make it easier to find the values of the variable.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have numerous real-world applications. By understanding how to solve quadratic equations and how to apply them in real-world situations, we can make predictions and solve problems in various fields.