Mathematical+Reflections,+64+0809

Platonic Solids NG Big Idea: Essential Question: Notes from Class: (4.2) Golden Rule of Algebra: What you do to one side, you must do to the other. The example below shows one way to solve the equation 100 + 4x = 25 + 7x

100 + 4x = 25 + 7x 100 + 4x - 4x = 25 + 7x - 4x 100 = 25 + 3x 100 - 25 = 25 + 3x - 25 75 = 3x 75/3 = 3x/3 25 = x (4.3) **F**irst **O**uter **I**nner **L**ast Although you can solve equations by making either tables or graphs, it's more efficient to use the symbolic method.

There are multiple ways to solve a linear equation. The most obvious way is to make tables and/or graphs and compare them to get your answer. However, you can also use the symbolic method. The symbolic method is when you erase all of the other operations in the equation until you're left with x and its value. For example, to solve 15=10+2x, this is what you would do: 15 = 10 + 2x 15-10 = 10-10 + 2x 5 = 2x 5/2 = 2x/2 2.5 = x
 * 1. Describe some general strategies for solving linear equations. Give examples that illustrate your strategies.**

One strategy to solve a quadratic equation of the form ax^2+bx=0 is to make a graph or table of it, and look for the x-intercepts. The x-intercepts, or roots, are the solution. Another strategy is to still look for the roots, but by turning the expanded equation into equivalent factored expressions instead of graphing it. The strategy of turning expanded form into factored form will definitely work. Take the equation x^2+6x+9=0. If you put it into factored form,it becomes (x+3)(x+3). This means that there is only one root, -3, so -3 is the solution.
 * 2. a. What general strategies can you use to solve quadratic equations of the form ax^2+bx=0?**
 * b. Will the strategies you described in part a work for solving quadratic equations of the form ax^2+bx+c=0? Use an example to help explain your answer.**

To solve mx+b=0 by using a graph, you could find the x-intercept, and that would be your solution. For example, the graph of y=5x+10 looks like this: If you look for the x-intercept, you'll see that is is -2. You've found your solution! An example of an ax+b=cx+d equation is 4x+8=3x+12. To solve these kinds of equations, you would look for the point of intersection between the two lines. In this case, the lines intersect at (4, 24). So 4 and 24 are the two possible answers for this equation. To solve ax^2+bx=0 by using a graph, you find the x-intercepts on the parabola---these are your answers. I graphed the equation 2x^2+5x. As you can see, the x-intercepts (roots) are -2.5 and 0---the two possible solutions.
 * 3. a. How could you solve a linear equation of the form //mx+b=0// by using a graph?**
 * b. How could you solve a linear equation of the form //ax+b=cx+d// by using a graph?**
 * c. How could you solve a quadratic equation of the form //ax^2+bx=0// by using a graph?**


 * Summary**: There are many possible strategies to use when solving linear and quadratic equations. These strategies involve graphs, tables, and sometimes just algebra. For linear equations, I prefer to use the symbolic method, but for quadratic equations I think using a graph is best.