Given a quadratic equation with the leading coefficient of 1, factor it.
Find two numbers whose product equals
c and whose sum equals
b .
Use those numbers to write two factors of the form
where
k is one of the numbers found in step 1. Use the numbers exactly as they are. In other words, if the two numbers are 1 and
the factors are
Solve using the zero-product property by setting each factor equal to zero and solving for the variable.
Factoring and solving a quadratic with leading coefficient of 1
Factor and solve the equation:
To factor
we look for two numbers whose product equals
and whose sum equals 1. Begin by looking at the possible factors of
The last pair,
sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors.
To solve this equation, we use the zero-product property. Set each factor equal to zero and solve.
The two solutions are
and
We can see how the solutions relate to the graph in
[link] . The solutions are the
x- intercepts of
Using the zero-product property to solve a quadratic equation written as the difference of squares
Solve the difference of squares equation using the zero-product property:
Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property.
Factoring and solving a quadratic equation of higher order
When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping, which requires four terms. With the equation in standard form, let’s review the grouping procedures:
With the quadratic in standard form,
multiply
Find two numbers whose product equals
and whose sum equals
Rewrite the equation replacing the
term with two terms using the numbers found in step 1 as coefficients of
x.
Factor the first two terms and then factor the last two terms. The expressions in parentheses must be exactly the same to use grouping.
Factor out the expression in parentheses.
Set the expressions equal to zero and solve for the variable.
Solving a quadratic equation using grouping
Use grouping to factor and solve the quadratic equation:
First, multiply
Then list the factors of
The only pair of factors that sums to
is
Rewrite the equation replacing the
b term,
with two terms using 3 and 12 as coefficients of
x . Factor the first two terms, and then factor the last two terms.
Astronomy (from Ancient Greek ἀστρονομία (astronomía) 'science that studies the laws of the stars') is a natural science that studies celestial objects and phenomena. It uses mathematics, physics, and chemistry in order to explain their origin and evolution.
Rafael
vjuvu
Elgoog
what is big bang theory?
Rosemary
what type of activity astronomer do?
Rosemary
No
Richard
the big bang theory is a theory which states that all matter was compressed together in one place the matter got so unstable it exploded releasing All its contents in the form of hydrogen
according to the theory of astronomers why the moon is always appear in an elliptical orbit?
Gatjuol
hi !!! I am new in astronomy....
I have so many questions in mind ....
all of scientists of the word they just give opinion only.
but they never think true or false ...
i respect all of them...
I believes whole universe depending
on true ...থিউরি
Govinda
hello
Jackson
hi
Elyana
we're all stars and galaxies a part of sun. how can science prove thx with respect old ancient times picture or books..or anything with respect to present time .but we r a part of that universe
there many theory to born universe but what is the reality of big bang theory to born universe
Asmit
what is the exact value of π?
Nagalakshmi
by big bang
universal
there are many theories regarding this it's on you believe any theory that you think is true ex. eternal inflation theory, oscillation model theory, multiple universe theory the big bang theory etc.
Aarya
I think after Big Bang!
Michele
from where on earth could u observe all the stars during the during the course of an year
is that so. the question was in the end of this chapter
Karuna
in theory, you could see them all from the equator (though over the course of a year, not at pne time). stars are measured in "declination", which is how far N or S of the equator (90* to -90*). Polaris is the North star, and is ALMOST 90* (+89*).
So it would just barely creep over the horizon.
Christopher
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