Solving equations
When we talk about solving equations we’re talking about determining whether a point rests on a line or not. When we can determine a range of values that rest on a line, curve or shape, we have solved the equation.
There are several strategies for solving equations. We have factorization where we pull the it apart and find simpler terms equal to the original.
Substitution, and elimination as well, compare two or more equations and apply Musa’s algorithm to find the range of values. What we do to one side we do to another. There’s the guess and check method, which is one way to resolve the equations from earlier. We found solutions for two of the three squared functions by guess and check. Though it still left us with one unresolved equation. We can resolve x squared minus x plus 1 using the quadratic formula.
These formulas are more complex. The quadratic formula is a method where a squared function is rearranged with the x value by itself on one side. It’s a messy equation that applies some creative algebra and factoring though it makes so we only have to plug in given information to find the range of solutions. And we find when we plug in one negative one and one for the a b and c values, we get 1 plus or minus i sq rt of three .. over two.
This introduces imaginary numbers which is another complexly elegant phenomenon of math. This term came to be from a series of math duels between several esteemed mathematicians in the 1600s. The imaginary plane and the solution to the cubic formula were each discovered as a result of the duels.
Basically a cubic formula was found to work for most circumstances. though in some cases, a negative number was left under a square root sign for which there is no answer because any number squared is always positive. So they came with with the term Italic I Indicating the square root of negative one. This made it possible to solve equations in these circumstances. I recommend “welshlabs” on YouTube for a great explanation of the imaginary plane and the amazing story about how this term came to be.
The cubic formula is even more complex. As was mentioned any cubic equation may be reduced to x cubed minus ax plus b. And when we plug in negative one and one for a and b, we get one half plus or minus sq rt 13 over six.
When we put in one half minus sq rt 13/6 we get negative .68. And when we put that into the cubic equation, we find it resolves the equation. In this case, the imaginary number was reduced to a negative one. Which makes the equation true. However when we put the result from the quadratic formula into the equation it doesn’t compute and I guess this is because of the i, the imaginary term.
Okay we covered a lot in this area. Thanks for hanging with me. Main things to remember here:
Equations show relationships.
They may be plotted as curves on a graph
There are several ways to find a range of values to make an equation true.
If a value makes an equation true, the value is on the curve.