How do we factor quadratic equations?

2 Answers
Mar 27, 2015

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.

Mar 27, 2015

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.