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Fill in the blank with a number to make the expression a perfect square.

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

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 Trinomial and also write its binomial equation.

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

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:

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

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

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 Trinomial 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:

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

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