Solve For \[$ S \$\].\[$(s+1)(s-5)=0\$\]Write Your Answers As Integers Or As Proper Or Improper Fractions In Simplest Form.\[$ S = \square \$\] Or \[$ S = \square \$\]
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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 quadratic equations of the form (s+1)(s-5)=0, where s is the variable. We will break down the solution process into manageable steps and provide examples to illustrate each step.
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. 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.
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
Solving the Given Equation
Now, let's apply the quadratic formula to the given equation (s+1)(s-5)=0. To do this, we need to expand the equation and rewrite it in the standard form ax^2 + bx + c = 0.
Step 1: Expand the Equation
(s+1)(s-5) = s^2 - 5s + s - 5 = s^2 - 4s - 5
Step 2: Rewrite the Equation in Standard Form
s^2 - 4s - 5 = 0
Step 3: Identify the Coefficients
a = 1, b = -4, c = -5
Step 4: Plug the Coefficients into the Quadratic Formula
s = (-(-4) ± √((-4)^2 - 4(1)(-5))) / 2(1) = (4 ± √(16 + 20)) / 2 = (4 ± √36) / 2 = (4 ± 6) / 2
Simplifying the Solutions
Now, we have two possible solutions:
s = (4 + 6) / 2 = 10 / 2 = 5
s = (4 - 6) / 2 = -2 / 2 = -1
Conclusion
In this article, we solved the quadratic equation (s+1)(s-5)=0 using the quadratic formula. We expanded the equation, rewrote it in standard form, identified the coefficients, and plugged them into the quadratic formula. Finally, we simplified the solutions to obtain the final answers.
Final Answers
The final answers are:
s = 5 s = -1
Tips and Tricks
- When solving quadratic equations, always start by expanding the equation and rewriting it in standard form.
- Identify the coefficients a, b, and c, and plug them into the quadratic formula.
- Simplify the solutions to obtain the final answers.
Common Mistakes to Avoid
- Failing to expand the equation and rewrite it in standard form.
- Failing to identify the coefficients a, b, and c.
- Failing to simplify the solutions.
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.
Conclusion
In conclusion, solving quadratic equations is a crucial skill for students and professionals alike. By following the steps outlined in this article, you can solve quadratic equations with ease. Remember to expand the equation, rewrite it in standard form, identify the coefficients, and simplify the solutions. With practice and patience, you will become proficient in solving quadratic equations and be able to apply them to real-world problems.
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Introduction
Quadratic equations can be a challenging topic 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: x = (-b ± √(b^2 - 4ac)) / 2a. Alternatively, you can factor the equation, if possible, or use the method of completing the square.
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: How do I apply the quadratic formula?
A: To apply the quadratic formula, you need to identify the coefficients a, b, and c, and plug them into the formula. Then, simplify the expression to obtain the solutions.
Q: What are the different types of solutions to a quadratic equation?
A: A quadratic equation can have three types of solutions:
- Two distinct real solutions
- One repeated real solution
- Two complex solutions
Q: How do I determine the number of solutions to a quadratic equation?
A: To determine the number of solutions to a quadratic equation, you can use the discriminant, which is the expression b^2 - 4ac. 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 two complex solutions.
Q: What is the discriminant?
A: The discriminant is the expression b^2 - 4ac, which is used to determine the number of solutions to a quadratic equation.
Q: How do I use the discriminant to determine the number of solutions?
A: To use the discriminant, you need to calculate the expression b^2 - 4ac. If the result is positive, the equation has two distinct real solutions. If the result is zero, the equation has one repeated real solution. If the result is negative, the equation has two complex solutions.
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 expand the equation and rewrite it in standard form
- Failing to identify the coefficients a, b, and c
- Failing to simplify the solutions
- Failing to use the correct formula or method
Q: How do I apply quadratic equations to real-world problems?
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.
Conclusion
In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the quadratic formula, the discriminant, and the different types of solutions, you can apply quadratic equations to a wide range of real-world problems. Remember to avoid common mistakes and practice solving quadratic equations to become proficient in this important mathematical concept.