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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.

Need help solving this problem (2/7)^-2
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_