What Would Be The Most Logical First Step For Solving This Quadratic Equation?$x^2 + 2x - 14 = 6$A. Add 14 To Both Sides B. Take The Square Root Of Both Sides C. Divide Both Sides By $x$ D. Subtract 6 From Both Sides
Understanding Quadratic Equations
Quadratic equations are a type of polynomial equation that involves a squared variable. They are commonly represented in the form of ax^2 + bx + c = 0, where a, b, and c are constants. In this article, we will focus on solving quadratic equations of the form x^2 + 2x - 14 = 6.
The Most Logical First Step
When solving quadratic equations, it is essential to follow a logical and systematic approach. The first step in solving the equation x^2 + 2x - 14 = 6 is to isolate the quadratic term on one side of the equation. This can be achieved by subtracting 6 from both sides of the equation.
Why Subtract 6 from Both Sides?
Subtracting 6 from both sides of the equation is the most logical first step because it allows us to isolate the quadratic term on one side of the equation. By doing so, we can then apply various techniques to solve for the value of x.
The Correct Answer
The correct answer is D. Subtract 6 from both sides.
Explanation
To solve the equation x^2 + 2x - 14 = 6, we need to isolate the quadratic term on one side of the equation. This can be achieved by subtracting 6 from both sides of the equation.
x^2 + 2x - 14 = 6
Subtract 6 from both sides:
x^2 + 2x - 20 = 0
Now that we have isolated the quadratic term, we can apply various techniques to solve for the value of x.
Techniques for Solving Quadratic Equations
There are several techniques for solving quadratic equations, including factoring, completing the square, and using the quadratic formula. In this article, we will focus on using the quadratic formula to solve the equation x^2 + 2x - 20 = 0.
The Quadratic Formula
The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is represented as:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = 2, and c = -20. Plugging these values into the quadratic formula, we get:
x = (-(2) ± √((2)^2 - 4(1)(-20))) / 2(1)
x = (-2 ± √(4 + 80)) / 2
x = (-2 ± √84) / 2
x = (-2 ± 2√21) / 2
x = -1 ± √21
Conclusion
In conclusion, the most logical first step for solving the quadratic equation x^2 + 2x - 14 = 6 is to subtract 6 from both sides of the equation. This allows us to isolate the quadratic term on one side of the equation, making it easier to apply various techniques to solve for the value of x. By using the quadratic formula, we can find the solutions to the equation x^2 + 2x - 20 = 0.
Frequently Asked Questions
- What is the first step in solving a quadratic equation?
- Why is it essential to isolate the quadratic term on one side of the equation?
- What are the different techniques for solving quadratic equations?
- How do I use the quadratic formula to solve a quadratic equation?
Answers
- The first step in solving a quadratic equation is to isolate the quadratic term on one side of the equation.
- It is essential to isolate the quadratic term on one side of the equation because it allows us to apply various techniques to solve for the value of x.
- The different techniques for solving quadratic equations include factoring, completing the square, and using the quadratic formula.
- To use the quadratic formula, plug the values of a, b, and c into the formula and simplify to find the solutions to the equation.
References
- "Quadratic Equations" by Math Open Reference
- "Solving Quadratic Equations" by Khan Academy
- "Quadratic Formula" by Wolfram MathWorld
Quadratic Equations Q&A ==========================
Frequently Asked Questions
Q: What is a quadratic equation?
A: A quadratic equation is a type of polynomial equation that involves a squared variable. It is commonly represented in the form of ax^2 + bx + c = 0, where a, b, and c are constants.
Q: What are the different types of quadratic equations?
A: There are several types of quadratic equations, including:
- Monic quadratic equations: These are quadratic equations where the coefficient of the squared term is 1. For example, x^2 + 2x + 1 = 0.
- Non-monic quadratic equations: These are quadratic equations where the coefficient of the squared term is not 1. For example, 2x^2 + 3x + 1 = 0.
- Linear quadratic equations: These are quadratic equations where the coefficient of the squared term is 0. For example, 2x + 1 = 0.
Q: How do I solve a quadratic equation?
A: There are several techniques for solving quadratic equations, including:
- Factoring: This involves expressing the quadratic equation as a product of two binomials. For example, x^2 + 4x + 4 = (x + 2)^2.
- Completing the square: This involves rewriting the quadratic equation in the form of (x + a)^2 = b. For example, x^2 + 4x + 4 = (x + 2)^2.
- Using the quadratic formula: This involves using the formula x = (-b ± √(b^2 - 4ac)) / 2a to find the solutions to the quadratic equation.
Q: What is the quadratic formula?
A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is represented as:
x = (-b ± √(b^2 - 4ac)) / 2a
Q: How do I use the quadratic formula?
A: To use the quadratic formula, plug the values of a, b, and c into the formula and simplify to find the solutions to the equation.
Q: What are the solutions to a quadratic equation?
A: The solutions to a quadratic equation are the values of x that satisfy the equation. These can be real or complex numbers.
Q: How do I determine the number of solutions to a quadratic equation?
A: The number of solutions to a quadratic equation can be determined by 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 real solution. If the discriminant is negative, the equation has no real solutions.
Q: What is the discriminant?
A: The discriminant is the expression b^2 - 4ac in the quadratic formula. It determines the number of solutions to the quadratic equation.
Q: How do I graph a quadratic equation?
A: To graph a quadratic equation, use the following steps:
- Find the x-intercepts: These are the points where the graph intersects the x-axis.
- Find the y-intercept: This is the point where the graph intersects the y-axis.
- Plot the points: Use the x-intercepts and y-intercept to plot the points on the graph.
- Draw the graph: Use the points to draw the graph of the quadratic equation.
Q: What are the applications of quadratic equations?
A: Quadratic equations have numerous applications in various fields, 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.
- Computer Science: Quadratic equations are used in algorithms and data structures, such as sorting and searching.
Q: How do I use quadratic equations in real-life situations?
A: Quadratic equations can be used in various real-life situations, such as:
- Designing and optimizing systems: Quadratic equations can be used to design and optimize systems, such as bridges and buildings.
- Modeling motion: Quadratic equations can be used to model the motion of objects under the influence of gravity.
- Solving problems: Quadratic equations can be used to solve problems, such as finding the maximum or minimum value of a function.
Q: What are some common mistakes to avoid when solving quadratic equations?
A: Some common mistakes to avoid when solving quadratic equations include:
- Not isolating the quadratic term: This can make it difficult to apply various techniques to solve for the value of x.
- Not using the correct formula: This can lead to incorrect solutions.
- Not checking the solutions: This can lead to incorrect solutions.
Q: How do I check the solutions to a quadratic equation?
A: To check the solutions to a quadratic equation, plug the values of x into the original equation and simplify to verify that the equation is true.