Solve For { W $} : : : { W^2 - 6w + 9 = 13 \}
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, w^2 - 6w + 9 = 13, to find the value of w. We will break down the solution into manageable steps, using algebraic techniques and mathematical concepts to arrive at the final answer.
Understanding Quadratic Equations
A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, w) 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 equation, w is the variable, and the coefficients are a = 1, b = -6, and c = 9 - 13 = -4.
Rearranging the Equation
To solve the equation w^2 - 6w + 9 = 13, we need to rearrange it to the standard form of a quadratic equation, which is:
w^2 - 6w + 9 - 13 = 0
Simplifying the equation, we get:
w^2 - 6w - 4 = 0
Using the 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 = (-b ± √(b^2 - 4ac)) / 2a
In our equation, a = 1, b = -6, and c = -4. Plugging these values into the quadratic formula, we get:
w = (6 ± √((-6)^2 - 4(1)(-4))) / 2(1)
Simplifying the Expression
Simplifying the expression inside the square root, we get:
w = (6 ± √(36 + 16)) / 2
w = (6 ± √52) / 2
Simplifying the Square Root
The square root of 52 can be simplified as:
√52 = √(4 × 13) = 2√13
Substituting this value back into the equation, we get:
w = (6 ± 2√13) / 2
Simplifying the Expression
Simplifying the expression, we get:
w = 3 ± √13
Conclusion
In this article, we solved the quadratic equation w^2 - 6w + 9 = 13 to find the value of w. We used algebraic techniques and mathematical concepts to arrive at the final answer, which is w = 3 ± √13. This solution demonstrates the importance of quadratic equations in mathematics and their applications in various fields.
Final Answer
The final answer is w = 3 ± √13.
Frequently Asked Questions
- 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.
- How do I solve a quadratic equation? To solve a quadratic equation, you can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by x = (-b ± √(b^2 - 4ac)) / 2a.
- What is the 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 = (-b ± √(b^2 - 4ac)) / 2a.
Additional Resources
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equation Solver
- Wolfram Alpha: Quadratic Equation Solver
Introduction
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 mathematical concept.
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: To solve a quadratic equation, you can use the quadratic formula, which states that for an equation of the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Alternatively, you can also use factoring, completing the square, or graphing to solve quadratic equations.
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 = (-b ± √(b^2 - 4ac)) / 2a
Q: What is the difference between a quadratic equation and a linear equation?
A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one. The general form of a linear equation is:
ax + b = 0
where a and b are constants, and x is the variable. Quadratic equations, on the other hand, have a degree of two, making them more complex and challenging to solve.
Q: Can I solve a quadratic equation by factoring?
A: Yes, you can solve a quadratic equation by factoring if it can be expressed as a product of two binomials. For example:
x^2 + 5x + 6 = (x + 3)(x + 2) = 0
In this case, you can set each factor equal to zero and solve for x.
Q: What is the significance of the discriminant in the quadratic formula?
A: The discriminant is the expression under the square root in the quadratic formula, which is:
b^2 - 4ac
The discriminant determines the nature of the solutions to the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one repeated real solution. If the discriminant is negative, the equation has no real solutions.
Q: Can I use a calculator to solve a quadratic equation?
A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic equation solver that can be used to find the solutions to a quadratic equation.
Q: What are some common mistakes to avoid when solving quadratic equations?
A: Some common mistakes to avoid when solving quadratic equations include:
- Not simplifying the expression under the square root
- Not checking the solutions for extraneous solutions
- Not using the correct formula or method for solving the equation
- Not checking the solutions for accuracy
Conclusion
Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task. By understanding the quadratic formula, factoring, completing the square, and graphing, you can solve quadratic equations with ease. Remember to avoid common mistakes and use a calculator when necessary to ensure accurate solutions.
Final Answer
The final answer is that quadratic equations are a powerful tool for solving mathematical problems, and with practice and patience, you can master the art of solving them.
Additional Resources
- Khan Academy: Quadratic Equations
- Mathway: Quadratic Equation Solver
- Wolfram Alpha: Quadratic Equation Solver
- Quadratic Equation Calculator: A free online calculator for solving quadratic equations.