Factoring a quadratic equation is the opposite of expanding an equation. FOILing (first, outer, inner, last) is the common technique used to do this.
Example of foiling: (x + 4) (x + 1) = x2 + 1x + 4x + 4, which simplifies to x2 + 5x + 4.
Because factoring is the opposite of foiling we are going to move backwards, but in a different way. The solution to a factoring problem is the beginning of a foiling problem.
Factoring Example:
x2 + 5x + 4 = 0
NOTE: your equation might not be equal to anything. It may appear as x2 + 5x + 4. In that case do the following steps as they appear but leave out the "= 0" from your answer because it's not there. Also do not do STEP 4 if it's not equal to zero.
STEP 1. x2 is equal to x multiplied by x, so separate the x's.
(x ) (x ) = 0
STEP 2. Look at the last number in the quadratic equation (4), take the multiples of that number (4) and see which set of multiples adds up the the middle number (5).
Multiples of 4: 2 and 2 (2 x 2 = 4), 4 and 1 (4 x 1 = 4).
2 + 2 = 4, 4 is not the middle # (5). 4 + 1 = 5!! Use 4 and 1.
(x 4) (x 1) = 0
STEP 3. Look at the sign (positive or negative) of the middle number you are trying to add up to. The signs of the 4 and 1 must fit too.
Middle number is a POSITIVE 5.
postive 4 + positive 1 = positive 5 <--- that's right!
but negative 4 + positive 1 = negative 3 <--- that's wrong!
negative 4 + negative 1 = negative 5 <--- that's wrong!
positive 4 + negative 1 = positive 3 <--- that's wrong!
(x + 4) (x + 1) << GRAND ANSWER!!! (unless equal to 0)
(x + 4) (x + 1) = 0
STEP 4. Sometimes the equation will not be set equal to anything. If that's the case, then (x + 4) (x + 1) is the correct answer. If it is set equal to zero then take each part of the new equation equal to zero and solve for x.
x+4 = 0 >> subtract 4 from both sides and you get x = -4
and
x + 1 = 0 >> subtract 1 from both sides and you get x = -1
x = -4 or x = -1 << GRAND ANSWER!!! :)
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Graphically might be the simplest answer.
Pros: There are many real life situations in which the relationship between two variables is quadratic rather than linear. So to solve these situations quadratic equations are necessary. There is a simple equation to solve any quadratic equation. Cons: Pupils who are still studying basic mathematics will not be told how to solve quadratic equations in some circumstances - when the solutions lie in the Complex field.
Graph the equation then find the x intercepts.
In general, quadratic equations have graphs that are parabolas. The quadratic formula tells us how to find the roots of a quadratic equations. If those roots are real, they are the x intercepts of the parabola.
Equations of the form z^4+az^2+a_0 are known as biquadratic equations. They are quartic equations. In general they can be solved by reducing them to a quadratic equation where x=z^2 is the variable. Then you can use the quadratic formula or factor. So plugging in x to the biquadratic giives us x^2+ax+a_0.