Question

5-64. copy and complete each of the diamond problems below.
the pattern used in the diamond problems is shown at right.
$\begin{array}{|c|c|} hline x & xy \\ hline x+y & y \\ hline end{array}$
a. $\begin{array}{|c|c|} hline & 0.32 \\ hline 1.2 & \\ hline end{array}$
b. $\begin{array}{|c|c|} hline & \\ hline 11 & \\ hline end{array}$
$\begin{array}{|c|} hline 25 \\ hline end{array}$
c. $\begin{array}{|c|c|} hline & \\ hline -9 & \\ hline end{array}$
$\begin{array}{|c|} hline -7 \\ hline end{array}$
d. $\begin{array}{|c|c|} hline & 234 \\ hline 31 & \\ hline end{array}$

Explanation:

Step1: Recall diamond problem rules

Top = product of left/right, bottom = sum of left/right.

Step2: Solve part a

Top: $0.32 \times 1.2 = 0.384$
Bottom: $0.32 + 1.2 = 1.52$

Step3: Solve part b

Find $x,y$: $x+y=25$, $xy=11$.
Roots of $t^2-25t+11=0$:
$x=\frac{25+\sqrt{625-44}}{2}=\frac{25+\sqrt{581}}{2}$, $y=\frac{25-\sqrt{581}}{2}$

Step4: Solve part c

Top: $(-9) \times (-7) = 63$
Bottom: $(-9) + (-7) = -16$

Step5: Solve part d

Find $x,y$: $x+y=31$, $xy=234$.
Roots of $t^2-31t+234=0$:
$t=\frac{31\pm\sqrt{961-936}}{2}=\frac{31\pm5}{2}$ → $x=18$, $y=13$

Answer:

a. Top: $\boldsymbol{0.384}$, Bottom: $\boldsymbol{1.52}$
b. Left/Right: $\boldsymbol{\frac{25+\sqrt{581}}{2}}$ and $\boldsymbol{\frac{25-\sqrt{581}}{2}}$
c. Top: $\boldsymbol{63}$, Bottom: $\boldsymbol{-16}$
d. Left/Right: $\boldsymbol{13}$ and $\boldsymbol{18}$