How do you factor quadratic equations?

Let us learn to factor quadratic equations in steps:

Step 1: Standard form of a quadratic equation

An equation can only be factored when it is written in the standard form. The standard form of a quadratic equation is as follows:

`ax^2 + bx + c = 0`

You should check to see if the given equation is in the standard form. If it is not, then you have to convert the equation into the standard form.

Step 2: Middle term of a quadratic equation

Once an equation is in the standard form, it is very easy to guess the middle term, because it is in the middle :-). But mathematically, the middle term in a quadratic equation is the term containing the variable 'x' without any exponent, and optionally, a coefficient (number) with x.

For example, in the following quadratic equation : `x^2 + x + 1` the middle term is x. However, in the quadratic equation `x^2 + 2x + 1 = 0`, the middle term is 2x.

Step 3: Splitting the middle term

The rule for splitting the middle term in a quadratic equation is that the product of the two parts into which the middle term is split should be equal to the product of the first and the last terms. Thus, For splitting the middle term of a quadratic equation, you first need to find the product of the first term and the last term. In a quadratic equation, the first term is the one have x^2 in it, whereas the last term is the number without a variable with it.

For example in the equation `2x^2 + 6x + 4 = 0`, the product of the first and the last term is `2x^2 * 4 = 8x^2`.

Now if we have to split the middle term, `6x`, it would be split into two parts in many ways:

`3x + 3x = 6x`
`5x + x = 6x`
`2x + 4x = 6x`

Out of the above three parts, only the one is correct in which the two parts multiply to give us `8x^2`. Only the last one meets this requirement, and thus, now the middle term has been properly split into two parts, that is we can write the equation like this:

`2x^2 + 2x + 4x + 4 = 0 `

Step 4: Factoring

Now we have converted the quadratic equation from being a trinomial to being a four term polynomial. We will group it into two parts by taking out the common factors like this:

`2x(x + 1) + 4(x + 1) = 0`

`(x + 1)(2x+ 4) = 0`

Step 5: Zero Product Rule

The Zero Product Rule says that if two things are multiplied and the result is zero, then either of the two things may equal zero. Thus, in the equation `(x + 1)(2x+ 4) = 0`, either `(x + 1) = 0` or `(2x + 4) = 0`.

If `(x + 1) = 0`, then `x = -1`, and

if `(2x + 4) = 0`, then `x = -4/2 = -2`

Thus, we obtain two values for x, -1 and -2.

The roots of the equation, or the solution of the equation, or even, the zeroes of the given equation are -1 and -2. 

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