Chapter 1.6, Problem 2DC

Solution

a.

To state:

How is $d$ related to $b$ in the general statement $x^2+bx+c={\left(x+d\right)}^2$ .

  $d=\frac{b}{2}$

Given:

The general statement $x^2+bx+c={\left(x+d\right)}^2$ .

Concepts Used:

Coefficients of like terms of two equal polynomials are equal.

Calculation:

Write the polynomials in the given general statement $x^2+bx+c={\left(x+d\right)}^2$ .

  $\begin{array}{l}{x}^2+bx+c={\left(x+d\right)}^2\\ x^2+bx+c=x^2+2dx+d^2\end{array}$

Equate the coefficients of like terms:

  $\begin{array}{l}b=2d\\ \frac{b}{2}=d\end{array}$

And

  $c=d^2$

Conclusion:

  $d=\frac{b}{2}$ is the relation between $d$ and $b$ .

b.

To state:

How is $c$ related to $d$ in the general statement $x^2+bx+c={\left(x+d\right)}^2$ .

  $c=d^2$

Given:

The general statement $x^2+bx+c={\left(x+d\right)}^2$ .

Known from the previous part:

  $c=d^2$

Concepts Used:

Coefficients of like terms of two equal polynomials are equal.

Conclusion:

From the result obtained in previous part, $c=d^2$ is the relation between $c$ and $d$ .

c.

To state:

How to obtain the numbers in the second column of the table obtained in previous question directly from the coefficients of $x$ in the expressions from the first column.

The numbers in the second column of the table are obtained by first halving the coefficients of $x$ in the expressions from the first column and then squaring the result.

Given:

The general statement $x^2+bx+c={\left(x+d\right)}^2$ .

From answer of previous question:

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

Known from the previous parts:

  $c=d^2$ and $d=\frac{b}{2}$ .

Concepts Used:

Substitution.

Comparison.

Observation:

Note that the numbers in the second column of the table represent $c$ from the given general statement $x^2+bx+c={\left(x+d\right)}^2$ . Also the coefficient of $x$ in the expressions from the first column represent $b$ . Thus, effectively it is required to find the relation between $c$ and $b$ .

Calculation:

Express $c$ in terms of $b$ using known relations $c=d^2$ and $d=\frac{b}{2}$ :

  $\begin{array}{l}c=d^2\\ c={\left(\frac{b}{2}\right)}^2\end{array}$

Thus, $c$ is the square of half of $b$ .

Conclusion:

The numbers in the second column of the table are obtained by first halving the coefficients of $x$ in the expressions from the first column and then squaring the result.