Question

How do we factor quadratic equations?

Answer 1

A quadratic expression is completely factorizable if and only if its discriminant is positive. Given a quadratic expression of the form `ax^2+bx+c`, the discriminant `\Delta` is defined as `b^2-4ac`.

If the discriminant is negative, the solving formula
`x_{1,2}=\frac{-b\pm \sqrt(\Delta)}{2a}` doesn't work with real numbers, because it would involve the square root of a negative number.

If the discriminant equals zero, the solving formula reduces to
`x_{1,2}=\frac{-b}{2a}`, i.e. `x_1=x_2`. Once the solutions are found, we can write `p(x)=(x-x_1)(x-x_2)`, where `p(x)` is the quadratic form and `x_1`,`x_2` are the solutions. Since in this case `x_1=x_2`, we have that `(x-x_1)(x-x_2)=(x-x_1)^2`, and this would be the factorization of the quadratic.

If the discriminant is positive the same formulas hold, and this time `p(x)=(x-x_1)(x-x_2)` is the final representation of the quadratic, since it cannot be further simplified, because the terms `(x-x_1)` and `(x-x_2)` are linear.

Answer 2

A quadratic equation is simply another way of solving a problem if the solution cannot be factored logically.

First we can start with some quick review:

Let’s say we have the equation `x^(2)+ 2x - 3` for example. This equation could be solved logically using the factors of the first and last terms.

To begin, we can state the factors of the first term, `x^(2)`. Imagine there’s an invisible 1 in front of the `x^(2)`, therefore the factors are 1, because only `1 * 1, or -1*-1` will multiply to get one. Then we can analyze the third term, `-3`. The factors of `-3` are either `1 * -3, or -1 * 3`.

Now we can check and see if any of the factors can combine in order to get a `+2`, the middle term (don’t worry about the x’s, those will carry over). Recall `1= -1, 1`, and `-3 = 1, -1, 3, -3`

From our factors we can use a -1 and a 3 to get +2. Therefore,
`(x+3)(x-1)=0` is our derived factorization. Then plug in the values to make the statement true, -3 or 1 will both result in an answer of 0 and our the possible values for x.

However , when the logical factorization seen above is not possible, we can plug our numbers into the quadratic equation .

`ax^(2)+bx+c` is the standard way we view an equation. Using the values from the equation above, `a= 1, b=2, and c=-3`.

After our a, b, and c values are found we can plug them into the actual quadratic equation.

`(-b+-sqrt(b^(2)-4ac))/(2a)`

Note : This equation may look intimidating, but as long as you follow factoring rules, you should have no problem. It’s totally normal to come out with an answer containing square roots.