Quadratic equations may be solved by graphing, using the quadratic formula, completing the square, and factoring. What are the pros and cons of each of these methods? When might each method be most appropriate? Which method do you prefer? Explain why.
Sometimes the factors just jump out at you , such as in
x^2 + 7x + 12 = 0
Usually, if the coefficients are single digits, it only takes a few seconds to see if it factors.
If the coefficients are 2 digits or more, I do a quick calculation of the determinant b^2 - 4ac
If that value is a perfect square, then I know the quadratic factors, but having found the discriminant, I am over half way there to get the answers, so I keep going with the formula anyway.
If the coefficient of x^2 is 1 and the middle term is even, I will always use completing the square over the formula.
It gives you the simplified radical solution, whereas the formula answer usually has to be simplified
e.g.
x^2 + 8x - 3 = 0
completing the square:
x^2 + 8x + 16 = 3 + 16
(x+4)^2 = 19
x+4 = ± √19
x = -4 ± √19
formula way:
x = (-8 ± √(64 - 4(1)(-3)) )/2
= (-8 ± √76)/2
= (-8 ± 2√19)/2
= -4 ± √19
Completing the square appears easier in this case.
1. Graphing:
Pros:
- Provides a visual representation of the quadratic equation.
- Can help understand the relationship between the equation and its solutions.
- Can quickly estimate the approximate values of the solutions.
Cons:
- It can be difficult to determine the exact solutions from a graph.
- Graphing can be time-consuming, especially for more complex equations.
- May not be accurate when dealing with irrational or complex solutions.
Appropriate use: Graphing is most suitable when you need a general idea of where the solutions are located or when the equation has integer solutions.
2. Quadratic formula:
Pros:
- Provides an exact formula for finding the solutions.
- Works for all quadratic equations, regardless of complexity.
- Can find both real and complex solutions.
Cons:
- Involves a lengthy calculation process.
- Requires careful attention to detail during calculation to avoid mistakes.
- May not be the most intuitive method for understanding the relationship between the equation and its solutions.
Appropriate use: The quadratic formula is a reliable method when you need to find the exact solutions of a quadratic equation, regardless of complexity. It is especially useful when decimal or fractional values need to be found.
3. Completing the square:
Pros:
- Provides an algebraic approach to solve quadratic equations.
- Can be used to rewrite the equation in vertex form, revealing the vertex coordinates.
- Can be helpful in understanding the transformation of the graph.
Cons:
- Requires careful manipulation and understanding of algebraic expressions.
- Not as efficient for solving multiple or complex equations.
- May involve fractions in the process, making calculation more challenging.
Appropriate use: Completing the square is most suitable when the main goal is to determine the vertex coordinates and the equation is easily manipulated.
4. Factoring:
Pros:
- Can be a quick method for solving simple quadratic equations.
- Provides a clear understanding of the relationship between the equation and its factors.
- Can help in identifying other possible algebraic relationships.
Cons:
- May not always be applicable, as not all quadratic equations are easily factorable.
- Can be time-consuming for complex equations.
- Factoring may involve working with fractions or irrational numbers.
Appropriate use: Factoring is most useful when the equation can be easily factored, especially when there is an evident common factor or when the equation is in a specific form that allows factoring, such as a difference of squares.
As an AI bot, I do not have personal preferences. The selection of the most appropriate method depends on the specific equation and the desired outcome. It is beneficial to be familiar with all methods to choose the most suitable one based on the given equation's complexity and characteristics.