Solve For $w$.$14w^2 - 41w - 3 = 0$Write Each Solution As An Integer, Proper Fraction, Or Improper Fraction In Simplest Form. If There Are Multiple Solutions, Separate Them With Commas.

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Introduction

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 a specific quadratic equation, 14w2−41w−3=014w^2 - 41w - 3 = 0, and provide a step-by-step guide on how to find the solutions.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

The Quadratic Formula

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

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation.

Solving the Quadratic Equation

Now, let's apply the quadratic formula to solve the equation 14w2−41w−3=014w^2 - 41w - 3 = 0. We have:

a=14a = 14, b=−41b = -41, and c=−3c = -3

Substituting these values into the quadratic formula, we get:

w=−(−41)±(−41)2−4(14)(−3)2(14)w = \frac{-(-41) \pm \sqrt{(-41)^2 - 4(14)(-3)}}{2(14)}

Simplifying the expression, we get:

w=41±1681+16828w = \frac{41 \pm \sqrt{1681 + 168}}{28}

w=41±184928w = \frac{41 \pm \sqrt{1849}}{28}

w=41±4328w = \frac{41 \pm 43}{28}

Now, we have two possible solutions:

w=41+4328w = \frac{41 + 43}{28}

w=41−4328w = \frac{41 - 43}{28}

Simplifying the first solution, we get:

w=8428w = \frac{84}{28}

w=3w = 3

Simplifying the second solution, we get:

w=−228w = \frac{-2}{28}

w=−114w = -\frac{1}{14}

Conclusion

In this article, we have solved the quadratic equation 14w2−41w−3=014w^2 - 41w - 3 = 0 using the quadratic formula. We have found two solutions, w=3w = 3 and w=−114w = -\frac{1}{14}. These solutions are in simplest form, and they are integers and proper fractions, respectively.

Tips and Tricks

  • When solving quadratic equations, make sure to simplify the expression before applying the quadratic formula.
  • Use the quadratic formula to find the solutions, and then simplify the expression to get the final answer.
  • Check your work by plugging the solutions back into the original equation to ensure that they are true.

Common Mistakes

  • Failing to simplify the expression before applying the quadratic formula.
  • Not checking the work by plugging the solutions back into the original equation.
  • Not using the quadratic formula to find the solutions.

Real-World Applications

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 economic systems and make predictions about future trends.

Conclusion

Introduction

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?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form ax2+bx+c=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation.

Q: What is the quadratic formula?

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

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

where aa, bb, and cc are the coefficients of the quadratic equation.

Q: How do I simplify the quadratic formula?

A: To simplify the quadratic formula, you can start by simplifying the expression under the square root:

b2−4acb^2 - 4ac

Then, you can simplify the expression by factoring or using other algebraic techniques.

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

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

  • Failing to simplify the expression before applying the quadratic formula.
  • Not checking the work by plugging the solutions back into the original equation.
  • Not using the quadratic formula to find the solutions.

Q: How do I check my work when solving quadratic equations?

A: To check your work when solving quadratic equations, you can plug the solutions back into the original equation to ensure that they are true.

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

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 economic systems and make predictions about future trends.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, you can use the quadratic formula to solve quadratic equations with complex coefficients. However, you will need to use complex numbers and algebraic techniques to simplify the expression.

Q: How do I solve quadratic equations with complex coefficients?

A: To solve quadratic equations with complex coefficients, you can use the quadratic formula and simplify the expression using complex numbers and algebraic techniques.

Conclusion

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 and simplifying the expression, we can find the solutions to quadratic equations. Remember to check your work and use the quadratic formula to find the solutions. With practice and patience, you will become proficient in solving quadratic equations and applying them to real-world problems.

Additional Resources

  • Khan Academy: Quadratic Equations
  • Mathway: Quadratic Equations
  • Wolfram Alpha: Quadratic Equations

Practice Problems

  • Solve the quadratic equation x2+5x+6=0x^2 + 5x + 6 = 0 using the quadratic formula.
  • Solve the quadratic equation x2−3x−4=0x^2 - 3x - 4 = 0 using the quadratic formula.
  • Solve the quadratic equation x2+2x−3=0x^2 + 2x - 3 = 0 using the quadratic formula.

Answer Key

  • x=−2x = -2 or x=−3x = -3
  • x=4x = 4 or x=−1x = -1
  • x=−1x = -1 or x=3x = 3