Please rewrite the written ones in your own words.
DQ 1.Post a response to the following: How do you know if a quadratic equation will have one, two, or no solutions? How do you find a quadratic equation if you are only given the solution? Is it possible to have different quadratic equations with the same solution? Explain. Provide your classmate’s with one or two solutions with which they must create a quadratic equation.
You can determine the number of solutions for a quadratic equation by evaluating the discriminant: b^2-4ac. If that is positive, there are two real solutions. If it's zero, there is one real solution. If it's negative, then there are no real solutions.
If you are given solutions "m" and "n", you can get a quadratic by plugging them into these binomials:
(x-m)(x-n) = 0
Then you can FOIL the expression, if you want.
Yes, two different quadratics can have the same solution. That will happen if they are constant multiples of each other. For example: x^2+x = 0 and -x^2-x = 0 will have the same solutions.
Here are a couple examples:
1 solution: 3
equation:
(x-3)(x-3) = x^2 - 6x + 9
2 solutions: 0 and 4:
(x-0)(x-4) = x^2 - 4x
DQ2. Post a response to the following: Quadratic equations can 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? Why?
Quadratic formula: In my opinion, this is the most general method, and likely the best overall. It will always work, and if you memorize the formula, there is no guessing about how to apply it. The formula allows you to find real and complex solutions.
Graphing: graphing the equation will only give valid results if the equation has real solutions. The solutions are located where the graph crosses the x axis. If the solutions are irrational or fractions with large denominators, this method will only be able to approximate the solutions. If you have a graphing calculator, this method is the quickest. If you don't have a calculator, it can be tedious and difficult to graph the equation, and you have to be very precise when you do it.
Completing the square: This is probably the most difficult method. I find it hardest to remember how to apply this method. Since the quadratic formula was derived from this method, I don't think there is a good reason to use completing the square when you have the formula (and are allowed to use it).
Factoring: this is probably the easiest method for solving an equation with integer solutions. If you can see how to split up the original equation into its factor pair, this is the quickest and allows you to solve the problem in one step.
DQ3.Examine the following equation: y = -2x^2 + 20x + 1
- What shape does the graph of this equation make? (Hint: You should be able to tell just by looking at the first term of the equation.)
- Does the graph open up or down (Hint: You should also be able to answer this just by looking at the first term of the equation.)
The shape is a parabola, since there is an x^2 term.
It opens downward, since the x^2 term has a negative coefficient.
DQ4. Find the coordinates of the midpoint for the following equation: y = -2x^2 + 20x + 1.
Your answer should be in the form (x-coordinate, y-coordinate).
(Hint: Find the x-coordinate by using the Midpoint Formula [-b / (2a)]. Then find the y-coordinate by plugging the x-coordinate value into the original equation and solving for “y.”)
-b/(2a) = -20/(2*-2) = -20/-4 = 5
Plug that in:
-2(5)^2 + 20(5) + 1
= -50 + 100 + 1
= 51
The coordinate is:
(5, 51)
CHECKPOINT MATH PROBLEMS:
Please make sure that the questions haven’t changed.
Number 1:
Sol: √5, -√5
Inter: (√5, 0), (-√5, 0)
Number 2:
7, 11
Number 3:
-1, 9
Number 4:
7+√59, 7-√59
Number 5:
-5, 3
Number 6:
Inter:
Number 7:
0.8
Number 8:
-6+4√2, -6-4√2
Number 9:
Number 10:
Inter:
Number 11:
6, 7
Number 12:
-2, 3
Number 13:
Approx = 0.405, -7.405
Number 14:
4
Number 15:
No solution
Number 16:
16, 30
Number 17:
First = 34.95
Second = 29.95
Number 18:
Choice C: sqrt(c^2-b^2)
Number 19:
Choice B: sqrt(17n/z)
Number 20:
One real
Number 21:
One real
Number 22:
x^2 + 4x + 3
Number 23:
x^2 – 16x + 64
Number 24:
x^2 – 8x + 16
Number 25:
Number 26: