Solve For $x$ In The Equation $x^2 - 10x + 25 = 35$.A. $ X = 5 ± 2 5 X = 5 \pm 2\sqrt{5} X = 5 ± 2 5 [/tex]B. $x = 5 \pm \sqrt{35}$C. $x = 10 \pm 2\sqrt{5}$D. $ X = 10 ± 35 X = 10 \pm \sqrt{35} X = 10 ± 35 [/tex]
Introduction
Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand the different methods and techniques used to solve them. In this article, we will focus on solving a quadratic equation of the form x^2 - 10x + 25 = 35. We will use algebraic methods to solve for x and provide the correct solution.
Understanding the Equation
The given equation is a quadratic equation in the form of ax^2 + bx + c = 0, where a = 1, b = -10, and c = 25 - 35 = -10. To solve for x, we need to isolate x on one side of the equation.
Rearranging the Equation
The first step in solving the equation is to rearrange it to the standard form of a quadratic equation. We can do this by subtracting 35 from both sides of the equation:
x^2 - 10x + 25 = 35
x^2 - 10x - 10 = 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 for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, a = 1, b = -10, and c = -10. Plugging these values into the quadratic formula, we get:
x = (10 ± √((-10)^2 - 4(1)(-10))) / 2(1)
x = (10 ± √(100 + 40)) / 2
x = (10 ± √140) / 2
x = (10 ± 2√35) / 2
x = 5 ± √35
Conclusion
In conclusion, the correct solution to the equation x^2 - 10x + 25 = 35 is x = 5 ± √35. This solution is obtained by using the quadratic formula and simplifying the expression.
Discussion
The quadratic formula is a powerful tool for solving quadratic equations. It can be used to solve equations of the form ax^2 + bx + c = 0, where a, b, and c are constants. The formula states that the solutions for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
In our case, we used the quadratic formula to solve the equation x^2 - 10x + 25 = 35. We plugged in the values of a, b, and c into the formula and simplified the expression to obtain the solution x = 5 ± √35.
Comparison with Other Options
Let's compare our solution with the other options provided:
A. x = 5 ± 2√5
B. x = 5 ± √35
C. x = 10 ± 2√5
D. x = 10 ± √35
Our solution, x = 5 ± √35, matches option B. This is the correct solution to the equation x^2 - 10x + 25 = 35.
Final Thoughts
Solving quadratic equations is an essential skill in mathematics. The quadratic formula is a powerful tool for solving quadratic equations, and it can be used to solve equations of the form ax^2 + bx + c = 0. In this article, we used the quadratic formula to solve the equation x^2 - 10x + 25 = 35 and obtained the correct solution x = 5 ± √35.
Introduction
In our previous article, we solved the quadratic equation x^2 - 10x + 25 = 35 using the quadratic formula. We obtained the correct solution x = 5 ± √35. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.
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 for x are given by:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula to solve a quadratic equation?
A: To use the quadratic formula, you need to plug in the values of a, b, and c into the formula. Then, simplify the expression to obtain the solutions for x.
Q: What are the values of a, b, and c in the equation x^2 - 10x + 25 = 35?
A: In the equation x^2 - 10x + 25 = 35, a = 1, b = -10, and c = -10.
Q: How do I simplify the expression in the quadratic formula?
A: To simplify the expression, you need to follow the order of operations (PEMDAS):
- Evaluate the expressions inside the parentheses.
- Evaluate any exponential expressions.
- Evaluate any multiplication and division expressions from left to right.
- Evaluate any addition and subtraction expressions from left to right.
Q: What is the correct solution to the equation x^2 - 10x + 25 = 35?
A: The correct solution to the equation x^2 - 10x + 25 = 35 is x = 5 ± √35.
Q: Why is the quadratic formula important?
A: The quadratic formula is important because it can be used to solve quadratic equations of the form ax^2 + bx + c = 0. This is a fundamental concept in mathematics, and it has many real-world applications.
Q: Can I use the quadratic formula to solve any quadratic equation?
A: Yes, you can use the quadratic formula to solve any quadratic equation of the form ax^2 + bx + c = 0.
Q: What are some common mistakes to avoid when using the quadratic formula?
A: Some common mistakes to avoid when using the quadratic formula include:
- Not plugging in the correct values of a, b, and c.
- Not simplifying the expression correctly.
- Not following the order of operations (PEMDAS).
Q: How can I practice using the quadratic formula?
A: You can practice using the quadratic formula by solving quadratic equations of the form ax^2 + bx + c = 0. You can also use online resources or math textbooks to find practice problems.
Q: What are some real-world applications of the quadratic formula?
A: The quadratic formula has many real-world applications, including:
- Physics: The quadratic formula is used to solve problems involving motion, energy, and momentum.
- Engineering: The quadratic formula is used to solve problems involving design, construction, and optimization.
- Economics: The quadratic formula is used to solve problems involving supply and demand, cost-benefit analysis, and optimization.
Conclusion
In conclusion, the quadratic formula is a powerful tool for solving quadratic equations. It can be used to solve equations of the form ax^2 + bx + c = 0, and it has many real-world applications. We hope that this Q&A section has helped to clarify any doubts or questions that readers may have.