To solve the problem of finding the value of \( c \) by completing the square, we need to follow a systematic approach. The general method for completing the square involves transforming a quadratic expression of the form \( x^2 + bx + c \) into a perfect square trinomial.
Steps to Complete the Square:
1.
Identify the coefficient of the linear term (\( b \)).
2.
Divide \( b \) by 2 and square the result to find the value that completes the square.
3.
Add this squared value to the expression to make it a perfect square trinomial.
Let's solve each problem step by step.
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Problem 1: \( S^2 + 13S + c \)
1. Identify \( b = 13 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{13}{2}\right)^2 = \frac{169}{4} \).
3. Therefore, \( c = \frac{169}{4} \).
Answer: \( c = \frac{169}{4} \)
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Problem 2: \( w^2 - 20w + c \)
1. Identify \( b = -20 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{-20}{2}\right)^2 = (-10)^2 = 100 \).
3. Therefore, \( c = 100 \).
Answer: \( c = 100 \)
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Problem 3: \( n^2 + 14n + c \)
1. Identify \( b = 14 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{14}{2}\right)^2 = 7^2 = 49 \).
3. Therefore, \( c = 49 \).
Answer: \( c = 49 \)
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Problem 4: \( y^2 + 3y + c \)
1. Identify \( b = 3 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{3}{2}\right)^2 = \frac{9}{4} \).
3. Therefore, \( c = \frac{9}{4} \).
Answer: \( c = \frac{9}{4} \)
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Problem 5: \( k^2 - 6k + c \)
1. Identify \( b = -6 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{-6}{2}\right)^2 = (-3)^2 = 9 \).
3. Therefore, \( c = 9 \).
Answer: \( c = 9 \)
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Problem 6: \( h^2 + 24h + c \)
1. Identify \( b = 24 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{24}{2}\right)^2 = 12^2 = 144 \).
3. Therefore, \( c = 144 \).
Answer: \( c = 144 \)
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Problem 7: \( s^2 - 15s + c \)
1. Identify \( b = -15 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{-15}{2}\right)^2 = \left(-\frac{15}{2}\right)^2 = \frac{225}{4} \).
3. Therefore, \( c = \frac{225}{4} \).
Answer: \( c = \frac{225}{4} \)
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Problem 8: \( b^2 + 22b + c \)
1. Identify \( b = 22 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{22}{2}\right)^2 = 11^2 = 121 \).
3. Therefore, \( c = 121 \).
Answer: \( c = 121 \)
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Problem 9: \( n^2 + 10n + c \)
1. Identify \( b = 10 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{10}{2}\right)^2 = 5^2 = 25 \).
3. Therefore, \( c = 25 \).
Answer: \( c = 25 \)
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Problem 10: \( p^2 - 16p + c \)
1. Identify \( b = -16 \).
2. Compute \( \left(\frac{b}{2}\right)^2 = \left(\frac{-16}{2}\right)^2 = (-8)^2 = 64 \).
3. Therefore, \( c = 64 \).
Answer: \( c = 64 \)
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Final Answers:
\[
\boxed{
\begin{array}{ll}
1) & c = \frac{169}{4} \\
2) & c = 100 \\
3) & c = 49 \\
4) & c = \frac{9}{4} \\
5) & c = 9 \\
6) & c = 144 \\
7) & c = \frac{225}{4} \\
8) & c = 121 \\
9) & c = 25 \\
10) & c = 64 \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of graphing quadratic functions worksheet answers algebra 2.
