4.3+&+4.4+Solve+by+Factoring

= 4.3 SOLVE BY FACTORING =

__Binomial__- Such as x+4, is the sum of two monomials. __Trinomial__- Such as x 2 +11x+28, is the sum of three monomials. __Quadratic Equation__- In one variable can be written in the form ax 2 +bx+c=0 where a doesn't equal 0. __Roots__-Solutions of a quadratic equation.
 * KEY VOCABULARY:** __Monomial__- An expression that is either a number, a variable, or the product of a number and one or more variables.

==// What is a quadratic equation? //== A quadratic equation is an equation that can be written in this form.ax2+bx+c=0 The a,b, and c here represent real number coefficients. So this means we are talking about an equation that is a constant times the variable squared plus a constant times the variable plus a constant equals zero, where the coefficient a on the variable squared can't be zero, because if it were then it would be a linear equation.
 * Quadratic Equations**

Examples
2x2+3x+1=0, x2+x=2x+3, (x+2)(x+3)=5 All these equations are equivalent to equations of the above form. The first one is already in that form. The second one can be put into it by subtracting 2x+3 from both sides. The third one can be put into it by multiplying out and then subtracting 5 from both sides.

// Standard Form //
The form ax2+bx+c=0 is the standard form for a quadratic equation, and for future reference, here the letter a will always mean the coefficient on the square of the variable, and b will be the coefficient on the variable, and c will be the constant term. To get a quadratic into standard form you must remove all parentheses and combine all like terms and add or subtract something from both sides so that the right side will be zero. Once you have your equation in standard form you can identify a,b, and c.

// Solving //
Now lets talk about solving these equations

Quadratic equations are harder to solve than linear equations, because once you have them in standard form it is hard to simplify them any further, and in this form there are still two occurrences of the variable, so it's hard to see what we can do to get the variable alone. So we have to find some clever tricks to get around this problem.

// Solving by Factoring //
One trick is to solve the equation by factoring. This trick works because of the principle of zero products. The principle of zero products says This is a very special property that only zero has. For other numbers there are lots of ways to multiply and get them, but not for zero. For zero, the only way to multiply numbers and get it, is if one of the numbers is zero.
 * If A and B are real numbers and AB=0, then either A=0 or B=0. **

The principle of zero products allows us to reduce a complicated equation to simpler equations provided the right side of the equation is zero, and the equation is factored, because we can set each of the factors equal to zero.


 * To solve a quadratic by factoring, first you must make sure it is in standard form. ** It is especially important that it is set equal to zero **, because remember, the principle of zero products only works for zero.
 * Then you must factor the left side.
 * Then you set each of your factors equal to zero and solve the equations you get to find the solutions to your equation.

// What if you can't factor? //
But some quadratics are difficult to factor, so for these equations we need other methods. The method of completing the square is a method that will work for any quadratic, but it is a little bit complicated, so I will introduce it slowly and step by step. But first to give you an overview of where we are going, I will show you a simple example of it.

Consider the following equation.

//**x2-2x-1=0**// This looks like a nice simple friendly equation, but we can't solve it by factoring, because we can't find two numbers to multiply and get -1 and add and get -2, so we are going to have to find another method.

But if only that minus sign on the 1 weren't there, then we could factor it really easily, in fact it would be a perfect square. How can we make that minus sign go away?

Well, one thing we could do is add 2 to both sides of the equation, and then the equation would become //**x2-2x+1=2**// and this factors to //**(x-1)2=2**//.

Now, I know what you're saying. You are saying, "But you said that you have to get it set equal to 0 to solve it by factoring, because the principle of zero products only works for 0. What good does it do to have something factored and set equal to 2?"

And if you are saying this to yourself, you are absolutely right. But this isn't just any old factorization. It is a ** perfect square **, and maybe you can do something with a perfect square set equal to 2.

If we could figure out how to solve equations like //**(x-1)2=2**// that is, perfect squares set equal to numbers, then we could solve an equation like //**x2-2x-1=0**//.

And if we could find a way to add a number to both sides of other quadratics so that we can put them into the form perfect square equals constant, then maybe we could be able to solve them too.

This means that to help us solve quadratic equations, we need to learn two skills. To work our way up to the task of solving equations of the form //**(x+k)2=d**// let's first start with the slightly easier task of solving equation of the form //**x2=d**//
 * # Solve equations of the form (x+k)2=d, where k and d are numbers.
 * 1) Find a way of figuring out what number to add to both sides of a quadratic equations so that the left side will become a perfect square. ||

Now let's look at the more general equation of the form //**(x+k)2=d**// This is really not much harder since anything you can do with x you should be able to do with x+k. x+k represents a number too. So solve for x+k and then add something to both sides of the equation to get x alone.

Example
Problem: Solve the equation. **[x - 2][x + 1] = 5** Solution: **x² - x - 2 = 5**
 * x² - x - 7 = 0**
 * a = 1, b = -1, c = -7**

// Completing the Square //
Now to problem number two, that of finding something to add to a quadratic to make it a perfect square.

This is what is meant by completing the square, and the secret to it is to expand out the expression //**(x+k)2**// and see what makes perfect squares tick. Applying our formula for squaring a binomial, we get //**(x+k)2=x2+2kx+k2**// The key here is to look at the relationship between the coefficient on x and the constant coefficient. The coefficient on x is 2k and the constant term is k2. This means that if we know the coefficient on x, and we want to know what the constant term has to be for the expression to be a perfect square, then we need to divide the coefficient on x by 2 to get k, and then square to get k2.

So if you have an expression of the form //**x2+bx**// and you want to find something to add to it to make it a perfect square, then you need to
 * # Divide b by 2 to get k
 * 1) Square k to get k2. ||

Example
Problem: Complete the square. Solution: 9/2 = 9/2, 9/2**² = 81/4** On the scratch paper you first divide the 9 by 2 and then square the result. Don't worry about the minus sign, because it will go away when you square anyway. Then the number you get will be the number you need to add to the expression to make a perfect square out of it. After you do that it is good practice to write it as the square that it is. For that you can use the first line of your scratch paper and match the sign with the sign of the second term of the original expression.
 * y² - 9y + 81/4 = (y - 9/2)²**

I hope the above has helped you understand the process of completing the square. If not, there is another approach to it that I have written an article about that you might find interesting for further understanding. It is a geometrical approach based on the method that many earlier mathematician used.

// Solving by Completing the Square //
Now we are ready to use the method of completing the squares to solve quadratic equations. The best way to do this is as follows. One thing we left out. So far all of the equations we have solved have had a coefficient of 1 on x2. What do we do if we have a coefficient other than 1 on there?
 * # Add something to both sides so that the left side has no constant term.
 * 1) Figure out what to add to the left side to make it a perfect square, and add that to both sides.
 * 2) Write the left side as the perfect square that it is and do the arithmetic on the right side.
 * 3) Solve the equation you get by the methods of equations of the form (x+k)2=d ||

Well, we don't really have any method of completing the squares to deal with that situation, so the easiest thing to do is just divide both sides by it and put up with the fractions. With completing the squares, fractions are not so bad to deal with because there is no guess work.

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 * INFORMITIVE VIDEO (WATCH): **

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Now that you have learned the method of completing the squares, I will tell you a secret. The methods of completing the squares is such a good method for solving quadratics that it is very seldom used for it. It is used for other things in mathematics. But for solving quadratic equations, it is such a good method that it puts itself out of business.You see, with such a mechanical method like the method of completing the squares, why not just apply it to the general quadratic equation and solve all quadratics in the world at once, and be done with it, and never have to use algebra to solve a quadratic again. == ==
 * || ==// The Quadratic Formula //==

Imaginary Solutions
This section is for more advanced students who know about imaginary numbers. If you know about imaginary numbers, you don't have to stop when you see the square root of a negative number, because with imaginary numbers you can take the square root of a negative number. To find the square root of any negative number you just take the square root of the corresponding positive number and multiply it by i, the square root of -1. This makes sense at least once you believe in the idea that the square root of -1 is i, because of the multiplication rule for square roots.

(It is customary usually to write the real number after the i when it is a square root so that it is clear that the i is not inside the radical.) Once you know how to find square roots of negative numbers, you find imaginary solutions to quadratics by the completing the square or the quadratic formula pretty much like you find real ones. For the following examples the instruction is to solve the equation. ||  ||

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