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In this section you will:
• Solve quadratic equations by factoring.
• Solve quadratic equations by the square root property.
• Solve quadratic equations by completing the square.

The computer monitor on the left in [link] is a 23.6-inch model and the one on the right is a 27-inch model. Proportionally, the monitors appear very similar. If there is a limited amount of space and we desire the largest monitor possible, how do we decide which one to choose? In this section, we will learn how to solve problems such as this using four different methods.

## Solving quadratic equations by factoring

An equation containing a second-degree polynomial is called a quadratic equation    . For example, equations such as $\text{\hspace{0.17em}}2{x}^{2}+3x-1=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{x}^{2}-4=0\text{\hspace{0.17em}}$ are quadratic equations. They are used in countless ways in the fields of engineering, architecture, finance, biological science, and, of course, mathematics.

Often the easiest method of solving a quadratic equation is factoring . Factoring means finding expressions that can be multiplied together to give the expression on one side of the equation.

If a quadratic equation can be factored, it is written as a product of linear terms. Solving by factoring depends on the zero-product property, which states that if $\text{\hspace{0.17em}}a\cdot b=0,$ then $\text{\hspace{0.17em}}a=0\text{\hspace{0.17em}}$ or $\text{\hspace{0.17em}}b=0,$ where a and b are real numbers or algebraic expressions. In other words, if the product of two numbers or two expressions equals zero, then one of the numbers or one of the expressions must equal zero because zero multiplied by anything equals zero.

Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression $\text{\hspace{0.17em}}\left(x-2\right)\left(x+3\right)\text{\hspace{0.17em}}$ by multiplying the two factors together.

$\begin{array}{ccc}\hfill \left(x-2\right)\left(x+3\right)& =& {x}^{2}+3x-2x-6\hfill \\ & =& {x}^{2}+x-6\hfill \end{array}$

The product is a quadratic expression. Set equal to zero, $\text{\hspace{0.17em}}{x}^{2}+x-6=0\text{\hspace{0.17em}}$ is a quadratic equation. If we were to factor the equation, we would get back the factors we multiplied.

The process of factoring a quadratic equation depends on the leading coefficient, whether it is 1 or another integer. We will look at both situations; but first, we want to confirm that the equation is written in standard form, $\text{\hspace{0.17em}}a{x}^{2}+bx+c=0,$ where a , b , and c are real numbers, and $\text{\hspace{0.17em}}a\ne 0.\text{\hspace{0.17em}}$ The equation $\text{\hspace{0.17em}}{x}^{2}+x-6=0\text{\hspace{0.17em}}$ is in standard form.

We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor    (GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section.

## The zero-product property and quadratic equations

The zero-product property    states

where a and b are real numbers or algebraic expressions.

A quadratic equation    is an equation containing a second-degree polynomial; for example

$a{x}^{2}+bx+c=0$

where a , b , and c are real numbers, and if $\text{\hspace{0.17em}}a\ne 0,$ it is in standard form.

In the quadratic equation $\text{\hspace{0.17em}}{x}^{2}+x-6=0,$ the leading coefficient, or the coefficient of $\text{\hspace{0.17em}}{x}^{2},$ is 1. We have one method of factoring quadratic equations in this form.

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey