For this week of Pre-Calculus 11, I learned about solving quadratic equations.
Quadratic equations are usually a trinomial or a factored form of a trinomial, that is equivalent to 0. In these types of equations, we aren’t only trying to factor the trinomial, rather find 2 values of x. While most equations only have 1 answer, quadratic equations have 2, as noted by its exponent.
Take a look above. A quadratic equation will always have an exponent on the x. This indicates that its factored form will have 2 xs, and therefor 2 different values. See how the x in (x + 2) must be -2 to make a zero pair, while the x in (x + 1) has to be -1. -2 and -1 are obviously different values, so there are 2 solutions to (x + 2)(x + 1) = 0.
On the other hand, the x in (x + 5 = 0) will always be -5. This is because the equation must equal 0; -5 + 5 = 0. The same thing for 8x + 4 = 0; the x in this equation must be (1/2). If we added any number, the equation will not equal to 0 and instead will be greater or lesser than 0, depending on the chosen value. These two equations only have 1 value, making them linear equations, not quadratic.
Sometimes, equations aren’t all that simple. A factored form of a trinomial might have coefficients alongside x, or even fractions or decimals. But, they’re just about the same as if you were trying to solve for a variable. Remember to isolate the variable, and the opposites of operations. Addition – subtraction, multiplication – division, squaring – square root, and changing decimals to fractions if needed.
Also notice that our quadratic equations always equal to 0. That is because, well, they must always equal to 0. Every, single time. This is due to the Zero Product Law: a rule which indicates that in an equation of (a * b = 0), one of the variables will always be 0. If the expression equaled to any other number, the Zero Product Law will not work and we wouldn’t be able to solve the quadratic equation, unless we take quick detour.
For equations where we have a trinomial in its entire or factored form equal to a number that is not 0, we can simply move the number on the right (or left, depending on where you want the 0 to be), to the opposite side so that one of the sides is 0. Or, if the other side is a factor of a trinomial, then we can just do FOIL to convert to its entire form, so we can properly move the other number to the trinomial’s side.
Now, we can factor our new trinomial to find the 2 values of x.
