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# How do you explain the zero product property and how it is used in solving quadratic equations?

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## Related Questions

###### Asked in Math and Arithmetic, Algebra, Geometry

### How many existing methods are there in solving quadratic equations?

There are 5 existing methods in solving quadratic equations. For
the first 4 methods (quadratic formula, factoring, graphing,
completing the square) you can easily find them in algebra books. I
would like to explain here the new one, the Diagonal Sum Method,
recently presented in book titled:"New methods for solving
quadratic equations and inequalities" (Trafford 2009). It directly
gives the 2 roots in the form of 2 fractions, without having to
factor the equation. The innovative concept of the method is
finding 2 fractions knowing their Sum (-b/a) and their Product
(c/a). It is very fast, convenient and is applicable whenever the
given quadratic equation is factorable. In general, it is hard to
tell in advance if a given quadratic equation can be factored.
However, if this new method fails to find the answer, then we can
conclude that the equation can not be factored, and consequently,
the quadratic formula must be used. This new method can replace the
trial-and-error factoring method since it is faster, more
convenient, with fewer permutations and fewer trials.

###### Asked in Math and Arithmetic, Algebra

### What is the zero product property and how do you use it?

Anything multiplied by zero is zero. And its converse, that the
product of non-zero numbers must be non-zero.
This is used to find the roots of equations.
If a function, f(x), is equal to the product of functions g(x)
and h(x),
that is, f(x) = g(x)*h(x)
then
f(x) = 0 implies that g(x) = 0 or h(x) = 0
At high school level, this is used to find the solutions of
quadratic equations:
If (x - a)*(x - b) = 0 then x - a = 0 or x - b = 0
that is, x = a or x = b

###### Asked in Algebra

### How do you solve quadratic equations by factoring?

There is a new method, called Diagonal Sum Method, that quickly
and directly give the 2 roots without having to factor the
equation. The innovative concept of this method is finding 2
fractions knowing their sum (-b/a) and their product (c/a). It is
fast, convenient and is applicable to any quadratic equation in
standard form ax^2 + bx + c = 0, whenever it can be factored. If it
fails to find answer, then the equation is not factorable, and
consequently, the quadratic formula must be used. So, I advise you
to proceed solving any quadratic equation in 2 steps. First, find
out if the equation can be factored? How?. Use this new method to
solve it. It usually takes fewer than 3 trials. If its fails then
use the quadratic formula to solve it in the second step. See book
titled:" New methods for solving quadratic equations and
inequalities" (Trafford Publishing 2009)

###### Asked in Math and Arithmetic, Algebra, Calculus

### What are two algebraic methods for solving quadratic equations?

Finally, there are two methods to use, depending on if the given
quadratic equation can be factored or not.
1.- The first one is the new Diagonal Sum Method, recently
presented in book titled: "New methods for solving quadratic
equations" (Trafford 2009). This method directly gives the two
roots in the form of two fractions, without having to factor it.
The innovative concept of this new method is finding 2 fractions
knowing their product (c/a) and their sum (-b/a). This new method
is applicable to any quadratic equation that can be factored. It
can replace the existing trial-and-error factoring method since
this last one contains too many more permutations. In general, it
is hard to tell in advance if a given quadratic equation can be
factored. However, if the new method fails to get the answers, then
you can positively conclude that this equation can not be factored.
Consequently, the quadratic formula must be used in solving. We
advise students to always try to solve the given equation by the
new method first.
If the student gets conversant with this method, it usually take
less than 2 trials to get answers.
2. the second one uses the quadratic formula that students can
find in any algebra book. This formula must be used for all
quadratic equations that can not be factored.

###### Asked in Algebra

### What is the special cases of quadratic equation?

The standard form of a quadratic equation is: ax^2 + bx + c = 0.
Depending on the values of the constants (a, b, and c), a quadratic
equation may have 2 real roots, one double roots, or no real
roots.
There are many "special cases" of quadratic equations.
1. When a = 1, the equation is in the form: x^2 + bx + c
= 0. Solving it becomes solving a popular puzzle: find 2 numbers
knowing their sum (-b) and their product (c). If you use the new
Diagonal Sum Method (Amazon e-book 2010), solving is fast and
simple.
Example: Solve x^2 + 33x - 108 = 0.
Solution. Roots have opposite signs. Write factor pairs of c =
-108. They are: (-1, 108),(-2, 54),(-3, 36)...This sum is -3 + 36 =
33 = -b. The 2 real roots are -3 and 36. There is no needs for
factoring.
2. Tips for solving 2 special cases of quadratic
equations.
a. When a + b + c = 0, one real root is (1) and the other
is (c/a).
Example: the equation 5x^2 - 7x + 2 = 0 has 2 real roots: 1 and
2/5
b. When a - b + c = 0, one real roots is (-1) and the
other is (-c/a)
Example: the equation 6x^2 - 3x - 9 = 0 has 2 real roots: (-1)
and (9/6).
3. Quadratic equations that can be factored.
The standard form of a quadratic equation is ax^2 + bx + c = 0.
When the Discriminant D = b^2 - 4ac is a perfect square, this
equation can be factored into 2 binomials in x: (mx + n)(px + q)=
0. Solving the quadratic equation results in solving these 2
binomials for x. Students should master how to use this factoring
method instead of boringly using the quadratic formula.
When a given quadratic equation can be factored, there are 2
best solving methods to choose:
a. The "factoring ac method" (You Tube) that determines
the values of the constants m, n, p, and q of the 2 above mentioned
binomials in x.
b. The Diagonal Sum Method (Amazon ebook 2010) that
directly obtains the 2 real roots without factoring. It is also
considered as "The c/a method", or the shortcut of the factoring
method. See the article titled" Solving quadratic equations by the
Diagonal Sum Method" on this website.
4. Quadratic equations that have 2 roots in the form of 2
complex numbers.
When the Discriminant D = b^2 - 4ac < 0, there are 2 roots in
the form of 2 complex numbers.
5. Some special forms of quadratic equations:
- quadratic equations with parameters: x^2 + mx - 7 + 0 (m is a
parameter)
- bi-quadratic equations: x^4 - 5x^2 + 4 = 0
- equations with rational expression: (ax + b)/(cx + d) = (ex +
f)
- equations with radical expressions.

###### Asked in Algebra

### How does factoring help solve quadratic equations?

Well, that's one method to solve the quadratic equation. Here is
an example (using the symbol "^" for power): solve x^2 - 5x + 6 =
0
Step 1: Convert the equation to a form in which the right side
is equal to zero. (Already done in this example.)
Step 2: Factor the left side. In this case, (x - 3) (x - 2) =
0
Step 3: Use the fact that if a product is zero, at least one of
its factors must be zero. This lets you convert the equation to two
equations;
x - 3 = 0 OR x - 2 = 0
Step 4: Solve each of the two equations.