Squared Trinomial

What is a Trinomial?

Trinomial is a Greek word composed of two components:

Tri - is three in Greek
Nom - is partition - with unknown to the square (that is, to the second power)

Therefore: The squared trinomial is an algebraic expression composed of three terms, one with the unknown to the square (to the power $2$ ), the second with the unknown without exponentiation, and the third, numbers without unknowns or letters that are different from those carrying the unknown.

For example, the form:

$X^2+6X+8$

Is a trinomial. Likewise, the form:

$-X^2-4X+8$

is also, even though some of the coefficients of the unknowns are negative.

And, in a general form:

$aX^2+bX+c$

It is a squared trinomial when the letters $a,b,c$ : are correct for any number we place in their place, except for $0$

Note that, placing a $0$ in place of the a will nullify the structure of the quadratic form
on the other hand, placing a $0$ in place of the parameters $b$ or $c$ would turn the trinomial expression (three monomials) into one with only $2$ monomials.

Quadratic expression breakdown showing that two numbers must multiply to constant term c and add to coefficient b in the standard form x² + bx + c

Detailed explanation

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Test yourself on solving trinomials!

$$ 4x^2=12x-9 $$

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Basic concepts related to the quadratic equation and the trinomial

Quadratic form

An algebraic expression with an unknown $X$ or $Y$ to the power of $2$ is the highest exponent

Coefficient

The coefficient is a number that is written to the left of the variable.

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Test your knowledge

Question 1

Solve the following problem:

$$ x^2+10x=-25 $$

Question 2

What is the value of x?

$$ x^4-x^3=2x^2 $$

Question 3

Solve the following problem:

$$ x^2=6x-9 $$

Parameter

A parameter is a letter other than $X$ or $Y$ that is found within the expression instead of a number

Monomial

Expression composed of the product or the quotient of a certain number with a variable such as: $X5$

Do you know what the answer is?

Question 1

Solve the following problem:

$$ x^2+144=24x $$

Question 2

$$ 60-16y+y^2=-4 $$

Question 3

Solve the following problem:

$$ x^2-10x=-16 $$

Polynomial

Expression composed of the addition or subtraction of $2$ or more monomials in which there are numbers and variables such as:

$X+3$

$4X-2$ and others like this

Where we will see squared trinomials in our math studies

In solving a quadratic equation without the formula
In simplifying algebraic fractions that require converting a trinomial into a multiplication of factors to simplify
In 4 mathematical operations with algebraic fractions to carry out the simplification or finding the appropriate common denominator, each trinomial structure is broken down into two factors

To this end, we will learn to break down every trinomial - a quadratic expression of three monomials into the multiplication of two polynomial factors.

Check your understanding

Question 1

Solve for y:

$$ y^2+4y+2=-2 $$

Question 2

$$ x^3=x^2+2x $$

Question 3

Solve for x:

$$ x^2+32x=-256 $$

Factoring Trinomials

To multiply the following expressions between parentheses let's return to the extended distributive property

$(X+3)(X+2)=$

We will obtain:

$X^2+3X+2X+2 \times 3=$

$X^2+(2+3)X+2 \times 3=$

$X^2+5X+6$

To perform the opposite operation from trinomial to the original given multiplication, we will look for $2$ numbers whose product gives said number without the $X$ .

In our case, it is $6$ and the sum of both numbers results in the coefficient $X$ , which is the middle monomial of the trinomial -> in this case $5$ .

$2\times3=6$

$5=2+3$

Then we factorize

$(X+2)(X+3)$

and we will look for two numbers that comply with

$aX^2+bX+c$

So, in the following trinomial:

$X^2+8X+12$

We will find the numbers whose product is $12$ and their sum $8$ . These are: $6$ and $2$ , since:

$8=2+6$

$12=6\times2$

Therefore, the factorization is:

$(X+2)(X+6)$

Also in the following form:

$X^2-12X+20$

We will find the numbers whose product is $20$ and their sum $12$ , knowing that both must be negative since the product is positive and the sum is negative.

Consequently, we will obtain:

$20=(10-)\times(2-)$

$12-=(10-)+(2-)$

Therefore, the factorization is:

$(X+2)(X+6)$

What happens when the product of the factors is negative?

This indicates that these numbers have different signs and, when the sum is positive, it also suggests that the number with the larger defined value is positive.

For example:

$X^2+5X-6$

The numbers $6$ and $(-1)$ , their sum is $5$ and their product $(-6)$ . Therefore, we will factorize:

$(X-1)(X+6)$

$X^2-7X-8$

Let's find $2$ numbers whose product is $(-8)$ and whose sum is $(-7)$ , and these are: $(-8)$ and $1$

Because: $8-=(8-)(1)$

and also: $7-=(8-)+1$

Therefore, the factorization is:

$(X+1)(X-8)$

Do you think you will be able to solve it?

Question 1

$$ 3x^3-10x^2+7x=0 $$

Question 2

$$ x^3+x^2-12x=0 $$

Question 3

$$ x^3-7x^2+6x=0 $$

How to Solve Quadratic Equations Without a Formula

$X^2+11X+24=0$

We will look for $2$ numbers whose product is $24$ and whose sum is $11$ , and these are: $8, 3$ and we will obtain:

$(X+8)(X+3)=0$

To get the result $0$ it could be that the value of the expression in parentheses is $0=(8+X)$

This will happen when $8 =X$

When $0= 3+X$

So $x=-3$

And, in this way, simplifying algebraic fractions, we will factorize every trinomial and we can consider various possibilities for simplification:

$\frac{x^2+2x+1}{x^2-x-2}=\frac{\left(x+1\right)\left(x+1\right)}{\left(x+1\right)\left(x-2\right)}$

and we will arrive at:

$\frac{x+1}{x-2}$

It cannot be simplified further due to the addition and subtraction operations between them.

Likewise, we can simplify in multiplication operations of fractions by decomposing the trinomial into a multiplication of two factors and simplifying as much as possible.

$\frac{1}{x-1}\times\frac{x^2-4x+3}{x-3}=$

$\frac{1}{x-1}\times\frac{\left(x-1\right)\left(x-3\right)}{x-3}=1$

Also in division operations: After converting the exercise to another of multiplicative inverse we will simplify as much as possible after factoring the trinomial.

$\frac{x^2+2x+1}{x^2-6+9}:\frac{x^2+3x+2}{x^2-x-6}$

$\frac{\left(x+1\right)\left(x+1\right)}{\left(x-3\right)\left(x-3\right)}\times\frac{\left(x-3\right)\left(x+2\right)}{\left(x+2\right)\left(x+1\right)}=\frac{x+1}{x-3}$

Similarly, to find the common denominator:

$\frac{1}{x^2-2x-3}-\frac{1}{x-3}$
$\frac{1}{\left(x-3\right)\left(x+1\right)}-\frac{1}{x-3}$
$\frac{1-\left(x+1\right)}{\left(x-3\right)\left(x+1\right)}=\frac{-x}{\left(x-3\right)\left(x+1\right)}$

How do you factor a trinomial in which the coefficient 2X is different from 1?

In general terms: $aX^2+bX+c$

We will look for two terms whose sum is $b$ and two factors whose product is $ac$

Term $A +$ Term $B = b$

Term $A×$ Term $B = ac$
We will break down $b$ into the sum of these two terms, which will allow us to factor the trinomial by taking the common factor out of the parentheses

For example:

$2X^2+5X+3=$

$3+2=5 ,~ 3×2=6$

Therefore, we will obtain:

$2X^2+2X+3X+3=$

$2X(X+1)+3(X+1)=$

$(2X+3)(X+1)$

Test your knowledge

Question 1

$$ 4x^2=12x-9 $$

Question 2

Solve the following problem:

$$ x^2+10x=-25 $$

Question 3

What is the value of x?

$$ x^4-x^3=2x^2 $$

Another system for factoring using the quadratic formula:

Thus, for example:

$2X^2+5X+3=$

We will obtain:

$2(X^2+2.5X+1.5)=$

With the quadratic formula we will find the reset numbers of the trinomial within the parentheses:

$X^2+2.5X+1.5=0$

$a=1,~b=2.5,~ c=1.5$

Now let's place it in the quadratic formula

$\frac{-b\pm\sqrt{b^2}-4ac}{2a}=$

and we will arrive at:

$X_1=-1$ , $X_2=-1.5$

a. We will start by factorizing $X_1$ and $X_2$ :

$a(X-X_1)(X-X_2)$

Therefore, the following factorization will be obtained:

$2(X+1.5)(X+1)$

We multiply the $2$ from the first parentheses to get whole numbers and it will be factorized:

$(2X+3)(X+1)$

Substitution of numbers for letters in the trinomial

When substituting numbers in a trinomial with letters, it is essential that we do it according to the basic rules.

If $1=a$ , we can carry out a shortened factorization

We will look for 2 terms whose product is $c$ and their sum $b$ .

For example:

$X^2+2ax-3a^2$

We will discover that the product of the expressions is:

$-a×3a$

$-3a^2$

and their sum is $2a$ , therefore, we will obtain

$(X+3a)(X-a)$

Also, when a is a number that is not $1$ , we will factorize the coefficient $X$ into two terms whose product equals $ac$ .

Thus, for example:

$2X^2+(2-a)X-a$

and we will obtain:

$2X^2+2X-aX-a$

$2X(X+1)-a(X+1)$

We will take out the common factor $X+1$ and obtain:

$(X+1)(2X-a)$

Examples and exercises with solutions of square trinomial

Exercise #1

$4x^2=12x-9$

Video Solution

Step-by-Step Solution

To solve the quadratic equation $4x^2 = 12x - 9$ , we begin by rewriting it in standard quadratic form:

$4x^2 - 12x + 9 = 0$

Here, we compare to the general form $ax^2 + bx + c = 0$ and identify:

$a = 4$
$b = -12$
$c = 9$

We will now use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Substitute in the values for $a$ , $b$ , and $c$ :

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4 \cdot 4 \cdot 9}}{2 \cdot 4}$

Simplify:

$x = \frac{12 \pm \sqrt{144 - 144}}{8}$

$x = \frac{12 \pm \sqrt{0}}{8}$

$x = \frac{12 \pm 0}{8}$

This simplifies further to:

$x = \frac{12}{8} = \frac{3}{2}$

Therefore, the solution to the equation $4x^2 = 12x - 9$ is $x = \frac{3}{2}$ .

Answer

$x=\frac{3}{2}$

Exercise #2

Solve the following problem:

$x^2+10x=-25$

Video Solution

Step-by-Step Solution

Proceed to solve the given equation:

$x^2+10x=-25$

First, let's arrange the equation by moving terms:

$x^2+10x=-25 \\ x^2+10x+25=0 \\$ Note that the expression on the left side can be factored using the perfect square trinomial formula for a binomial squared:

$(\textcolor{red}{a}+\textcolor{blue}{b})^2=\textcolor{red}{a}^2+2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

As shown below:

$25=5^2$

Therefore, we'll represent the rightmost term as a squared term:

$x^2+10x+25=0 \\ \downarrow\\ \textcolor{red}{x}^2+10x+\textcolor{blue}{5}^2=0$

Now let's examine again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}+\textcolor{blue}{b})^2=\textcolor{red}{a}^2+\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation that we obtained in the last step:

$\textcolor{red}{x}^2+\underline{10x}+\textcolor{blue}{5}^2=0$

Notice that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{5}^2$ indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor this expression (on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined with a line):

$(\textcolor{red}{a}+\textcolor{blue}{b})^2=\textcolor{red}{a}^2+\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we will query whether we can represent the expression on the left side as:

$\textcolor{red}{x}^2+\underline{10x}+\textcolor{blue}{5}^2=0 \\ \updownarrow\text{?}\\ \textcolor{red}{x}^2+\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{5}}+\textcolor{blue}{5}^2=0$

And indeed it is true that:

$2\cdot x\cdot5=10x$

Therefore we can represent the expression on the left side of the equation as a perfect square binomial:

$\textcolor{red}{x}^2+2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{5}+\textcolor{blue}{5}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{5})^2=0$

From here we can take the square root of both sides of the equation (and don't forget there are two possibilities - positive and negative when taking the square root of an even power), then we'll easily solve by isolating the variable:

$(x+5)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x+5=\pm0\\ x+5=0\\ \boxed{x=-5}$

Let's summarize the solution of the equation:

$x^2+10x=-25 \\ x^2+10x+25=0 \\ \downarrow\\ \textcolor{red}{x}^2+2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{5}+\textcolor{blue}{5}^2=0 \\ \downarrow\\ (\textcolor{red}{x}+\textcolor{blue}{5})^2=0 \\ \downarrow\\ x+5=0\\ \downarrow\\ \boxed{x=-5}$

Therefore the correct answer is answer C.

Answer

$x=-5$

Exercise #3

What is the value of x?

$x^4-x^3=2x^2$

Video Solution

Step-by-Step Solution

To solve the problem $x^4 - x^3 = 2x^2$ , let's proceed as follows:

Step 1: Set the equation to zero.
$x^4 - x^3 - 2x^2 = 0$
Step 2: Factor out the greatest common factor.
The common factor among all terms is $x^2$ .
Factoring out $x^2$ gives:
$x^2(x^2 - x - 2) = 0$
Step 3: Solve the factors.
This equation breaks into two factors that can be solved separately:
- $x^2 = 0$
- $x^2 - x - 2 = 0$
Step 4: Solve $x^2 = 0$ .
Since $x^2 = 0$ , we get:
$x = 0$
Step 5: Solve $x^2 - x - 2 = 0$ .
This can be factored further. We look for two numbers that multiply to $-2$ and add up to $-1$ .
These numbers are $-2$ and $1$ , so we factor as:
$(x - 2)(x + 1) = 0$
Step 6: Solve the quadratic factors.
Set each factor equal to zero:
- $x - 2 = 0 \Rightarrow x = 2$
- $x + 1 = 0 \Rightarrow x = -1$

The solutions to the equation $x^4 - x^3 = 2x^2$ are $x = -1, 0, 2$ .

Therefore, the correct answer is:

$x = -1, 2, 0$

Answer

$x=-1,2,0$

Exercise #4

Solve the following problem:

$x^2=6x-9$

Video Solution

Step-by-Step Solution

Proceed to solve the given equation:

$x^2=6x-9$

First, let's arrange the equation by moving terms:

$x^2=6x-9 \\ x^2-6x+9=0$

Note that we can factor the expression on the left side by using the perfect square trinomial formula for a binomial squared:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

As shown below:

$9=3^2$ Therefore, we'll represent the rightmost term as a squared term:

$x^2-6x+9=0 \\ \downarrow\\ \textcolor{red}{x}^2-6x+\textcolor{blue}{3}^2=0$

Now let's examine again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation that we obtained in the last step:

$\textcolor{red}{x}^2-\underline{6x}+\textcolor{blue}{3}^2=0$

Note that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{3}^2$ indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor the expression in question (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined with a single line):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we will query whether we can represent the expression on the left side of the equation as:

$\textcolor{red}{x}^2-\underline{6x}+\textcolor{blue}{3}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}}+\textcolor{blue}{3}^2=0$

And indeed it is true that:

$2\cdot x\cdot3=6x$

Therefore we can represent the expression on the left side of the equation as a perfect square binomial:

$\textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}}+\textcolor{blue}{3}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{3})^2=0$

From here we can take the square root of both sides of the equation (and don't forget that there are two possibilities - positive and negative when taking an even root of both sides of an equation), then we'll easily solve by isolating the variable:

$(x-3)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-3=\pm0\\ x-3=0\\ \boxed{x=3}$

Let's summarize the solution of the equation:

$x^2=6x-9 \\ x^2-6x+9=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{3}+\textcolor{blue}{3}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{3})^2=0 \\ \downarrow\\ x-3=0\\ \downarrow\\ \boxed{x=3}$

Therefore the correct answer is answer C.

Answer

$x=3$

Exercise #5

Solve the following problem:

$x^2+144=24x$

Video Solution

Step-by-Step Solution

Proceed to solve the given equation:

$x^2+144=24x$

Arrange the equation by moving terms:

$x^2+144=24x \\ x^2-24x+144=0$

Note that we are able to factor the expression on the left side by using the perfect square trinomial formula:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-2\textcolor{red}{a}\textcolor{blue}{b}+\textcolor{blue}{b}^2$

As demonstrated below:

$144=12^2$

Therefore, we'll represent the rightmost term as a squared term:

$x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-24x+\textcolor{blue}{12}^2=0$

Now let's examine once again the perfect square trinomial formula mentioned earlier:

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

And the expression on the left side in the equation that we obtained in the last step:

$\textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0$

Note that the terms $\textcolor{red}{x}^2,\hspace{6pt}\textcolor{blue}{12}^2$ indeed match the form of the first and third terms in the perfect square trinomial formula (which are highlighted in red and blue),

However, in order to factor this expression (which is on the left side of the equation) using the perfect square trinomial formula mentioned, the remaining term must also match the formula, meaning the middle term in the expression (underlined):

$(\textcolor{red}{a}-\textcolor{blue}{b})^2=\textcolor{red}{a}^2-\underline{2\textcolor{red}{a}\textcolor{blue}{b}}+\textcolor{blue}{b}^2$

In other words - we'll query whether we can represent the expression on the left side of the equation as:

$\textcolor{red}{x}^2-\underline{24x}+\textcolor{blue}{12}^2=0\\ \updownarrow\text{?}\\ \textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0$

And indeed it is true that:

$2\cdot x\cdot12=24x$

Therefore we can represent the expression on the left side of the equation as a perfect square trinomial:

$\textcolor{red}{x}^2-\underline{2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0$

$(x-12)^2=0\hspace{8pt}\text{/}\sqrt{\hspace{6pt}}\\ x-12=\pm0\\ x-12=0\\ \boxed{x=12}$

Let's summarize the solution of the equation:

$x^2+144=24x \\ x^2-24x+144=0 \\ \downarrow\\ \textcolor{red}{x}^2-2\cdot\textcolor{red}{x}\cdot\textcolor{blue}{12}+\textcolor{blue}{12}^2=0 \\ \downarrow\\ (\textcolor{red}{x}-\textcolor{blue}{12})^2=0 \\ \downarrow\\ x-12=0\\ \downarrow\\ \boxed{x=12}$

Therefore the correct answer is answer C.

Answer

$x=12$

Do you know what the answer is?

Question 1

Solve the following problem:

$$ x^2=6x-9 $$

Question 2

Solve the following problem:

$$ x^2+144=24x $$

Question 3

$$ 60-16y+y^2=-4 $$

Start practice