is (4+x)^2 the same as (x+4)^2

Yes, they are the same, as you can switch the numbers around within the parenthesis, just not outside of them.

Just curious, what's the point of doing this to an expression? Like, where will this be applied?

Pretty much all of algebra will be used when we start doing calculus which is where math gets really cool with way more real world applications like launching satellites into space and making vaccines. Endless possibilities! But we must have a SOLID foundation in algebra for all this cool stuff!

I don't understand question 4 and 5. in question 4 he said ""notice that the middle term is two times the product of the numbers that are squared: 2(3x)(5)=30x2(3x})(\D 5)=30x2(3x)(5)=30x2, l, 3, x, D, 5, equals, 30, x. then he does the equation again and shows the answer I don't understand how he got there

normally when you have for example x^2+6x+9 you would take the root of 9 and multiply it by 2. if the answer is the first-degree variable coefficient (6) then it satisfy the perfect squares "trick" rule, and the answer would be (x+3)^2. when you have a coefficient to the 2nd-degree variable like in 9x^2+30x+25, you do almost the same: first you find the root of the 0-degree variable (25), which is 5. (5^2=25) then you find the root of the 2nd-degree variable's coefficient (9), which is 3 (3^2=9). now here is the different from before: you need to multiply both roots (3 and 5) and then multiply them by 2 (as you normally would). if the answer is the same as the 1st-degree variable's coefficient ((5*3)*2=30) then the trick can be applied here also, giving you the answer made of those 2 roots you found erlier(3 and 5)-squared. (3x+5)^2 hope you understood and that it helped you.

Number three marks (X+7) squared wrong as well, but the "I need help" section is correct.

Maybe your Caps Lock is on. (X+7) is different of (x+7). Hope I have helped :)

is algebra 1 a high school class because im taking it in middle school

It can be, many people take algebra in middle school but there are also plenty of people who take it in high school. I think most people take geometry freshman year in high school (9th grade) but there are plenty of people who take it before or after

problem 7 doesnt have power (degree) button to write the correct answer. Need Fix

No fix is needed. You can use the ^ symbols to enter an exponent. (3x+4y)^2 Alternatively, you can enter it as (3x+4y)(3x+4y)

Main content

Course: Algebra 1 > Unit 13

Lesson 8: Factoring quadratics with perfect squares

Factoring quadratics: Perfect squares

Google Classroom

Learn how to factor quadratics that have the "perfect square" form. For example, write x²+6x+9 as (x+3)².

Factoring a polynomial involves writing it as a product of two or more polynomials. It reverses the process of polynomial multiplication.

In this article, we'll learn how to factor perfect square trinomials using special patterns. This reverses the process of squaring a binomial, so you'll want to understand that completely before proceeding.

Intro: Factoring perfect square trinomials

To expand any binomial, we can apply one of the following patterns.

$(a + b)^{2} = a^{2} + 2 a b + b^{2}$ ‍
$(a - b)^{2} = a^{2} - 2 a b + b^{2}$ ‍

Note that in the patterns,

a

and

b

can be any algebraic expression. For example, suppose we want to expand

(x + 5)^{2}

. In this case,

a = x

and

b = 5

, and so we get:

$\begin{aligned} (x + 5)^{2} & = x^{2} + 2 (x) (5) + (5)^{2} \\ = x^{2} + 10 x + 25 \end{aligned}$ ‍

You can check this pattern by using multiplication to expand

(x + 5)^{2}

The reverse of this expansion process is a form of factoring. If we rewrite the equations in the reverse order, we will have patterns for factoring polynomials of the form

a^{2} \pm 2 a b + b^{2}

$a^{2} + 2 a b + b^{2} = (a + b)^{2}$ ‍
$a^{2} - 2 a b + b^{2} = (a - b)^{2}$ ‍

We can apply the first pattern to factor

x^{2} + 10 x + 25

. Here we have

a = x

and

b = 5

$\begin{aligned} x^{2} + 10 x + 25 & = x^{2} + 2 (x) (5) + (5)^{2} \\ = (x + 5)^{2} \end{aligned}$ ‍

Expressions of this form are called perfect square trinomials. The name reflects the fact that this type of three termed polynomial can be expressed as a perfect square!

Let's take a look at a few examples in which we factor perfect square trinomials using this pattern.

Example 1: Factoring $x^{2} + 8 x + 16$ ‍

Notice that both the first and last terms are perfect squares:

x^{2} = (x)^{2}

and

16 = (4)^{2}

. Additionally, notice that the middle term is two times the product of the numbers that are squared:

2 (x) (4) = 8 x

This tells us that the polynomial is a perfect square trinomial, and so we can use the following factoring pattern.

$a^{2} + 2 a b + b^{2} = (a + b)^{2}$ ‍

In our case,

a = x

and

b = 4

. We can factor our polynomial as follows:

\begin{aligned} x^{2} + 8 x + 16 & = (x)^{2} + 2 (x) (4) + (4)^{2} \\ = (x + 4)^{2} \end{aligned}

We can check our work by expanding

(x + 4)^{2}

\begin{aligned} (x + 4)^{2} & = (x)^{2} + 2 (x) (4) + (4)^{2} \\ = x^{2} + 8 x + 16 \end{aligned}

Check your understanding

1) Factor $x^{2} + 6 x + 9$ ‍.

2) Factor $x^{2} - 6 x + 9$ ‍.

Notice that both the first and last terms are perfect squares:

x^{2} = (x)^{2}

and

9 = (3)^{2}

. Additionally, notice that the middle term is two times the product of the numbers that are squared:

2 (x) (3) = 6 x

This means that the polynomial is a perfect square trinomial. We can factor it as follows:

\begin{aligned} x^{2} - 6 x + 9 & = (x)^{2} - 2 (x) (3) + (3)^{2} \\ = (x - 3)^{2} \end{aligned}

Notice that the sign in the binomial is negative since the coefficient of the

x

-term is negative.

3) Factor $x^{2} + 14 x + 49$ ‍.

Example 2: Factoring $4 x^{2} + 12 x + 9$ ‍

It is not necessary for the leading coefficient of a perfect square trinomial to be

1

For example, in

4 x^{2} + 12 x + 9

, notice that both the first and last terms are perfect squares:

4 x^{2} = (2 x)^{2}

and

9 = (3)^{2}

. Additionally, notice that the middle term is two times the product of the numbers that are squared:

2 (2 x) (3) = 12 x

Because it satisfies the above conditions,

4 x^{2} + 12 x + 9

is also a perfect square trinomial. We can again apply the following factoring pattern.

$a^{2} + 2 a b + b^{2} = (a + b)^{2}$ ‍

In this case,

a = 2 x

and

b = 3

. The polynomial factors as follows:

\begin{aligned} 4 x^{2} + 12 x + 9 & = (2 x)^{2} + 2 (2 x) (3) + (3)^{2} \\ = (2 x + 3)^{2} \end{aligned}

We can check our work by expanding

(2 x + 3)^{2}

Yes! We can also factor

4 x^{2} + 12 x + 9

using the grouping method.

Let's start this process by finding factors of

4 \cdot 9 = 36

that add up to

12

. These two numbers are

6

and

6

since

6 \cdot 6 = 36

and

6 + 6 = 12

We can now break the term

12 x

into

6 x + 6 x

and use grouping to factor the polynomial:

\begin{aligned} 4 x^{2} + 12 x + 9 \\ = 4 x^{2} + 6 x + 6 x + 9 \\ = (4 x^{2} + 6 x) + (6 x + 9) & Group terms \\ = 2 x (2 x + 3) + 3 (2 x + 3) & Factor out GCFs \\ = 2 x (2 x + 3) + 3 (2 x + 3) & Common factor! \\ = (2 x + 3) (2 x + 3) & Factor out 2 x + 3 \\ = (2 x + 3)^{2} \end{aligned}

Notice, however, that recognizing

4 x^{2} + 12 x + 9

as a perfect square trinomial can save you some time!

Check your understanding

4) Factor $9 x^{2} + 30 x + 25$ ‍.

5) Factor $4 x^{2} - 20 x + 25$ ‍.

Notice that both the first and last terms are perfect squares:

4 x^{2} = (2 x)^{2}

and

25 = (5)^{2}

. Additionally, notice that the middle term is two times the product of the numbers that are squared:

2 (2 x) (5) = 20 x

This means that the polynomial is a perfect square trinomial. We can factor it as follows:

\begin{aligned} 4 x^{2} - 20 x + 25 & = (2 x)^{2} - 2 (2 x) (5) + (5)^{2} \\ = (2 x - 5)^{2} \end{aligned}

Notice that the sign in the binomial is negative since the coefficient of the

x

-term is negative.

Challenge problems

6*) Factor $x^{4} + 2 x^{2} + 1$ ‍.

7*) Factor $9 x^{2} + 24 x y + 16 y^{2}$ ‍.

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Aaishahali
Posted 6 years ago. Direct link to Aaishahali's post “is (4+x)^2 the same as (...”
is (4+x)^2 the same as (x+4)^2
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- Rex Schmitz 🐉🐍
  Posted 6 years ago. Direct link to Rex Schmitz 🐉🐍's post “Yes, they are the same, a...”
  Yes, they are the same, as you can switch the numbers around within the parenthesis, just not outside of them.
  Comment on Rex Schmitz 🐉🐍's post “Yes, they are the same, a...”
  (37 votes)
Sanjna Ayyar
Posted 7 years ago. Direct link to Sanjna Ayyar's post “Just curious, what's the ...”
Just curious, what's the point of doing this to an expression? Like, where will this be applied?
Button navigates to signup pageComment on Sanjna Ayyar's post “Just curious, what's the ...”
(11 votes)
Answer
- Theo Mostert
  Posted 4 years ago. Direct link to Theo Mostert's post “Pretty much all of algebr...”
  Pretty much all of algebra will be used when we start doing calculus which is where math gets really cool with way more real world applications like launching satellites into space and making vaccines. Endless possibilities! But we must have a SOLID foundation in algebra for all this cool stuff!
  Comment on Theo Mostert's post “Pretty much all of algebr...”
  (38 votes)
sean cabo
Posted 7 years ago. Direct link to sean cabo's post “Number three marks (X+7) ...”
Number three marks (X+7) squared wrong as well, but the "I need help" section is correct.
Button navigates to signup pageButton navigates to signup page
(4 votes)
Answer
- Gustavo Delazeri
  Posted 7 years ago. Direct link to Gustavo Delazeri's post “Maybe your Caps Lock is o...”
  Maybe your Caps Lock is on. (X+7) is different of (x+7).
  Hope I have helped :)
  Comment on Gustavo Delazeri's post “Maybe your Caps Lock is o...”
  (26 votes)
mtchuenkam
Posted 7 years ago. Direct link to mtchuenkam's post “I don't understand questi...”
I don't understand question 4 and 5. in question 4 he said ""notice that the middle term is two times the product of the numbers that are squared: 2(3x)(5)=30x2(3x})(\D 5)=30x2(3x)(5)=30x2, l, 3, x, D, 5, equals, 30, x. then he does the equation again and shows the answer I don't understand how he got there
Button navigates to signup pageComment on mtchuenkam's post “I don't understand questi...”
(7 votes)
Answer
- Eric Parker
  Posted 6 years ago. Direct link to Eric Parker's post “normally when you have fo...”
  normally when you have for example x^2+6x+9 you would take the root of 9 and multiply it by 2. if the answer is the first-degree variable coefficient (6) then it satisfy the perfect squares "trick" rule, and the answer would be (x+3)^2.
  
  when you have a coefficient to the 2nd-degree variable like in 9x^2+30x+25, you do almost the same:
  first you find the root of the 0-degree variable (25), which is 5. (5^2=25)
  then you find the root of the 2nd-degree variable's coefficient (9), which is 3 (3^2=9).
  
  now here is the different from before: you need to multiply both roots (3 and 5) and then multiply them by 2 (as you normally would). if the answer is the same as the 1st-degree variable's coefficient ((5*3)*2=30) then the trick can be applied here also, giving you the answer made of those 2 roots you found erlier(3 and 5)-squared.
  (3x+5)^2
  hope you understood and that it helped you.
  Button navigates to signup page
  (11 votes)
Ian Philbrick-Miller
Posted 7 years ago. Direct link to Ian Philbrick-Miller's post “What do the a and b stand...”
What do the a and b stand for in the equation a^2+2ab+b^2 ? I'm very confused :/
Button navigates to signup pageComment on Ian Philbrick-Miller's post “What do the a and b stand...”
(4 votes)
Answer
- Sans
  Posted 4 years ago. Direct link to Sans's post “they are just any varible...”
  they are just any varibles. a and b can be anything (a has to be equal to a and b has to be equal to b tho)
  Button navigates to signup page
  (1 vote)
HermioneGranger1219
Posted 4 years ago. Direct link to HermioneGranger1219's post “is algebra 1 a high schoo...”
is algebra 1 a high school class because im taking it in middle school
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(3 votes)
Answer
- KC
  Posted 4 years ago. Direct link to KC's post “It can be, many people ta...”
  It can be, many people take algebra in middle school but there are also plenty of people who take it in high school. I think most people take geometry freshman year in high school (9th grade) but there are plenty of people who take it before or after
  Comment on KC's post “It can be, many people ta...”
  (5 votes)
smitters
Posted a year ago. Direct link to smitters's post “problem 7 doesnt have pow...”
problem 7 doesnt have power (degree) button to write the correct answer. Need Fix
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(3 votes)
Answer
- Kim Seidel
  Posted a year ago. Direct link to Kim Seidel's post “No fix is needed. You ca...”
  No fix is needed. You can use the ^ symbols to enter an exponent. (3x+4y)^2
  
  Alternatively, you can enter it as (3x+4y)(3x+4y)
  Comment on Kim Seidel's post “No fix is needed. You ca...”
  (5 votes)
silly goofy goober
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so like do we do the stuff we learned before or ?? what are we doing??
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Loyal Brave And True
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Does anyone understand this at all?
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Owen Childers
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47 - The answer to the secret of the universe
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- FreeRadical
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Algebra 1

Course: Algebra 1 > Unit 13

Intro: Factoring perfect square trinomials

Example 1: Factoring x2+8x+16‍

Check your understanding

Example 2: Factoring 4x2+12x+9‍

Check your understanding

Challenge problems

Want to join the conversation?

Example 1: Factoring $x^{2} + 8 x + 16$ ‍

Example 2: Factoring $4 x^{2} + 12 x + 9$ ‍