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Math  /  Algebra

QuestionMiranda organized some of the expressions used so far. She has lined up each area expression, ax2+bx+ca x^{2}+b x+c, with the corresponding dimension expression, (x+h)2(x+h)^{2}. \begin{tabular}{|c|c|c|c|} \hline Area Expression & x2+8x+16x^{2}+8 x+16 & x2+6x+9x^{2}+6 x+9 & x2−10x+25x^{2}-10 x+25 \\ \hline Dimension Expression & (x+4)2(x+4)^{2} & (x+3)2(x+3)^{2} & (x−5)2(x-5)^{2} \\ \hline \end{tabular}
3. Miranda believes she has noticed a pattern and can determine hh using the value of bb. is there a pattern? If there is a pattern, describe the pattern and show how the value of bb can be used to determine the value of hh.
4. Miranda believes there is another pattern. She thinks the value of hh can be used to find the value of cc. Is there a pattern? If there is a pattern, describe the pattern and show how the value of hh can be used to determine the value of cc.
5. Miranda begins to wonder if bb can be used to determine the value of cc. Describe how bb can be used to determine the value of cc, and give one example to show how the value of bb can be used to determine the value of cc.

Studdy Solution

STEP 1

1. The expressions are quadratic polynomials.
2. The dimension expressions are perfect square trinomials.
3. We need to identify patterns relating b b , h h , and c c .

STEP 2

1. Identify the pattern between b b and h h .
2. Identify the pattern between h h and c c .
3. Explore the relationship between b b and c c .

STEP 3

Examine the given expressions and compare the coefficients:
- For x2+8x+16 x^2 + 8x + 16 and (x+4)2 (x+4)^2 , b=8 b = 8 and h=4 h = 4 . - For x2+6x+9 x^2 + 6x + 9 and (x+3)2 (x+3)^2 , b=6 b = 6 and h=3 h = 3 . - For x2−10x+25 x^2 - 10x + 25 and (x−5)2 (x-5)^2 , b=−10 b = -10 and h=−5 h = -5 .
Notice that b=2h b = 2h .

STEP 4

Using the relationship b=2h b = 2h , we can find h h from b b as h=b2 h = \frac{b}{2} .
Now, examine the relationship between h h and c c :
- For h=4 h = 4 , c=16=42 c = 16 = 4^2 . - For h=3 h = 3 , c=9=32 c = 9 = 3^2 . - For h=−5 h = -5 , c=25=(−5)2 c = 25 = (-5)^2 .
Notice that c=h2 c = h^2 .

STEP 5

Using the relationships b=2h b = 2h and c=h2 c = h^2 , we can express c c in terms of b b :
c=(b2)2=b24 c = \left(\frac{b}{2}\right)^2 = \frac{b^2}{4}
Example: If b=8 b = 8 , then c=824=644=16 c = \frac{8^2}{4} = \frac{64}{4} = 16 .
The patterns are:
1. h=b2 h = \frac{b}{2}
2. c=h2 c = h^2
3. c=b24 c = \frac{b^2}{4}

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