\[x^2-6x+?\]

The aim of this article is to find the **number** that when placed in the **blank** of the given **equation**, makes the equation expression a **perfect square**.

The basic concept behind this article is the **Perfect Square Trinomial**.

**Perfect Square Trinomials** are **quadratic polynomial equations** calculated by solving the **square** of the **binomials equation**. The solution involves the **factorization** of a given **binomial**.

A **Perfect Square Trinomial** is expressed as follows:

\[a^2x^2\pm2axb+b^2\]

Where:

$a$ and $b$ are the **roots of the equation**.

We can identify the **binomial equation** from the given **perfect square trinomial** as per the following steps:

$1.$ Check the **first** and **third terms** of the given **trinomial** if they are a **perfect square**.

$2.$ **Multiply** the **roots** $a$ and $b$.

$3.$ Compare the **product of the roots** $a$ and $b$ with the **middle term of trinomial**.

$4.$ If the **coefficient** of the **middle term** is equal to** two times** the **product of the square root** of the **first** and **third term** and the **first** and **third term** are **perfect square**, the given expression is proved to be a **Perfect Square Trinomial**.

This **Perfect Square Trinomial** is actually a solution of the **square** of a given **binomial** as follows:

\[\left(ax\pm b\right)^2=(ax\pm b)(ax\pm b)\]

Solving it as follows:

\[\left(ax\pm b\right)^2={(ax)}^2\pm(ax)(b)+{(\pm b)}^2\pm(b)(ax)\]

\[\left(ax\pm b\right)^2=a^2x^2\pm 2axb+b^2\]

## Expert Answer

The given expression is:

\[x^2-6x+?\]

We have to find the** third term** of the given **trinomial equation**, making it a **Perfect Square Trinomial**.

Let’s compare it with the **standard form** of **Perfect Square Trinomial**.

\[a^2x^2\pm2axb+b^2\]

By comparing the **first term** of the expressions, we know that:

\[a^2x^2=x^2\]

\[a^2x^2={{(1)}^2x}^2\]

Hence:

\[a^2=1\]

\[a=1\]

By comparing the **middle term** of the expressions, we know that:

\[2axb=6x\]

We can write it as follows:

\[2axb=6x=2(1)x(3)\]

Hence:

\[b=3\]

By comparing the **third term** of the expressions, we know that:

\[b^2=?\]

As we know:

\[b=3\]

So:

\[b^2=9\]

Hence:

\[a^2x^2\pm2axb+b^2={(1)x}^2-2(1)x(3)+{(3)}^2\]

And our **Perfect Square Trinomial** is as follows:

\[x^2-6x+9\]

And the **third term** of the **Perfect Square Trinomial** is:

\[b^2=9\]

For proof, its **binomial expression** can be expressed as follows:

\[\left(ax\pm b\right)^2={(x-3)}^2\]

\[{(x-3)}^2=(x-3)(x-3)\]

\[{(x-3)}^2={(x)}^2+(x)(-3)+(-3)(x)+(-3)(-3)\]

\[{(x-3)}^2=x^2-3x-3x+9\]

\[{(x-3)}^2=x^2-6x+9\]

## Numerical Result

The **third term** that makes the given expression a **Perfect Square Trinomial** is:

\[b^2=9\]

And our **Perfect Square Trinomial** is as follows:

\[x^2-6x+9\]

## Example

Find the** third term** of the given **Perfect Square Trinomia**l and also write its binomial equation.

\[4x^2+32x+?\]

We have to find the **third term** of the given **trinomial equatio**n, making it a **Perfect Square Trinomial**.

Let’s compare it with the standard form of **Perfect Square Trinomial**.

\[a^2x^2\pm2axb+b^2\]

By comparing the **first term** of the expressions, we know that:

\[a^2x^2={4x}^2\]

\[a^2x^2={{(2)}^2x}^2\]

Hence:

\[a^2={(2)}^2\]

\[a=2\]

By comparing the **middle term** of the expressions, we know that:

\[2axb=32x\]

We can write it as follows:

\[2axb=6x=2(2)x(8)\]

Hence:

\[b=8\]

By comparing the** third term** of the expressions, we know that:

\[b^2=?\]

As we know:

\[b=8\]

So:

\[b^2=64\]

Hence:

\[a^2x^2\pm2axb+b^2={(2)x}^2+2(2)x(8)+{(8)}^2\]

And our **Perfect Square Trinom**ial is as follows:

\[x^2+32x+64\]

And the **third term** of the **Perfect Square Trinomial** is:

\[b^2=64\]

Its **binomial expression** can be expressed as follows:

\[\left(ax\pm b\right)^2={(2x+8)}^2\]