Chapter 1.6, Problem 1DC

Solution

To fill: The black for the given table.

The obtained table is shown as follow,

Completing the square
Expression	Numbers of 1-tiles needed to complete the square	Expression written as square
$x^2+2x+\underset{¯}{?}$	1	${\left(x+1\right)}^2$
$x^2+4x+\underset{¯}{?}$	4	${\left(x+2\right)}^2$
$x^2+6x+\underset{¯}{?}$	9	${\left(x+3\right)}^2$
$x^2+8x+\underset{¯}{?}$	16	${\left(x+4\right)}^2$
$x^2+10x+\underset{¯}{?}$	25	${\left(x+5\right)}^2$

Given Information:

The table is defined as,

Completing the square
Expression	Numbers of 1-tiles needed to complete the square	Expression written as square
$x^2+2x+\underset{¯}{?}$	?	?
$x^2+4x+\underset{¯}{?}$	?	?
$x^2+6x+\underset{¯}{?}$	9	${\left(x+3\right)}^2$
$x^2+8x+\underset{¯}{?}$	?	?
$x^2+10x+\underset{¯}{?}$	?	?

Calculation:

Consider the given table,

First, find the number for the polynomial $x^2+2x+\underset{¯}{?}$ .

The general form of the perfect square is $a^2+2ab+b^2={\left(a+b\right)}^2$ .

Rewrite the given polynomial.

  $x^2+2x+\underset{¯}{?}=x^2+2\cdot x\cdot 1+\underset{¯}{?}$

Compare the above polynomial with the general form of the perfect square $a^2+2ab+b^2={\left(a+b\right)}^2$ , this implies that the value of $b$ is 1. And the missing number is 1.

  $\begin{array}{c}{x}^2+2x+1=x^2+2\cdot x\cdot 1+1\\ ={\left(x+1\right)}^2\end{array}$

Second, find the number for the polynomial $x^2+4x+\underset{¯}{?}$ .

The general form of the perfect square is $a^2+2ab+b^2={\left(a+b\right)}^2$ .

Rewrite the given polynomial.

  $x^2+4x+\underset{¯}{?}=x^2+2\cdot x\cdot 2+\underset{¯}{?}$

Compare the above polynomial with the general form of the perfect square $a^2+2ab+b^2={\left(a+b\right)}^2$ , this implies that the value of $b$ is 2. And the missing number is 4.

  $\begin{array}{c}{x}^2+4x+4=x^2+2\cdot x\cdot 2+4\\ ={\left(x+2\right)}^2\end{array}$

Fourth, find the number for the polynomial $x^2+8x+\underset{¯}{?}$ .

The general form of the perfect square is $a^2+2ab+b^2={\left(a+b\right)}^2$ .

Rewrite the given polynomial.

  $x^2+8x+\underset{¯}{?}=x^2+2\cdot x\cdot 4+\underset{¯}{?}$

Compare the above polynomial with the general form of the perfect square $a^2+2ab+b^2={\left(a+b\right)}^2$ , this implies that the value of $b$ is 4. And the missing number is 16.

  $\begin{array}{c}{x}^2+8x+4=x^2+2\cdot x\cdot 4+16\\ ={\left(x+4\right)}^2\end{array}$

Fifth, find the number for the polynomial $x^2+10x+\underset{¯}{?}$ .

The general form of the perfect square is $a^2+2ab+b^2={\left(a+b\right)}^2$ .

Rewrite the given polynomial.

  $x^2+10x+\underset{¯}{?}=x^2+2\cdot x\cdot 5+\underset{¯}{?}$

Compare the above polynomial with the general form of the perfect square $a^2+2ab+b^2={\left(a+b\right)}^2$ , this implies that the value of $b$ is 5. And the missing number is 25.

  $\begin{array}{c}{x}^2+10x+25=x^2+2\cdot x\cdot 5+25\\ ={\left(x+25\right)}^2\end{array}$

Substitute all the values in the table.

Completing the square
Expression	Numbers of 1-tiles needed to complete the square	Expression written as square
$x^2+2x+\underset{¯}{?}$	1	${\left(x+1\right)}^2$
$x^2+4x+\underset{¯}{?}$	4	${\left(x+2\right)}^2$
$x^2+6x+\underset{¯}{?}$	9	${\left(x+3\right)}^2$
$x^2+8x+\underset{¯}{?}$	16	${\left(x+4\right)}^2$
$x^2+10x+\underset{¯}{?}$	25	${\left(x+5\right)}^2$

Therefore, the solution of the given equation is $x=-3,-4$ .