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Home » \div 7 = 7 \text remainder 6 \Rightarrow 55 \equiv 6 \pmod7 - Crosslake

\div 7 = 7 \text remainder 6 \Rightarrow 55 \equiv 6 \pmod7 - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: Factor the divisor: $ x^2 - 3x + 2 = (x-1)(x-2) $. Use the Remainder Theorem: Let $ R(x) = ax + b $ be the remainder. Evaluate $ f(1) = 1 - 2 + 1 + 1 = 1 = a(1) + b $. Evaluate $ f(2) = 32 - 16 + 4 + 1 = 21 = a(2) + b $. Solve: $ a + b = 1 $, $ 2a + b = 21 $. Subtract: $ a = 20 $, then $ b = -19 $. Remainder: $ 20x - 19 $. \boxed{20x - 19}

📌 Question: If $ h(2y + 1) = y^3 - 5y + 6 $, find $ h(y^2 - 1) $.

📌 Solution: Let $ t = 2y + 1 \Rightarrow y = \frac{t - 1}{2} $. Substitute into $ h(t) $: $ h(t) = \left(\frac{t - 1}{2}\right)^3 - 5\left(\frac{t - 1}{2}\right) + 6 $. Simplify: $ h(t) = \frac{(t-1)^3}{8} - \frac{5(t-1)}{2} + 6 $. Now replace $ t $ with $ y^2 - 1 $: $ h(y^2 - 1) = \frac{(y^2 - 2)^3}{8} - \frac{5(y^2 - 2)}{2} + 6 $. Expand and combine terms (if required). \boxed{\frac{(y^2 - 2)^3}{8} - \frac{5(y^2 - 2)}{2} + 6}

📌 Question: Define $ M(v) = v - \frac{v^5}{5} $. If $ b_1 = 1 $ and $ b_{n+1} = M(b_n) $, compute $ b_3 $.

📌 Solution: Compute $ b_1 = 1 $. Then $ b_2 = 1 - \frac{1^5}{5} = \frac{4}{5} $. Next, $ b_3 = \frac{4}{5} - \frac{\left(\frac{4}{5}\right)^5}{5} = \frac{4}{5} - \frac{1024}{3125} = \frac{2500 - 1024}{3125} = \frac{1476}{3125} $. Simplify: $ \frac{1476}{3125} $ (already reduced). \boxed{\dfrac{1476}{3125}}

📌 Question: Determine the center of the hyperbola $ 4x^2 - 9y^2 + 8x + 18y - 11 = 0 $.

📌 Solution: Group and complete the square: $ 4(x^2 + 2x) - 9(y^2 - 2y) = 11 $. Complete squares: $ 4[(x+1)^2 - 1] - 9[(y-1)^2 - 1] = 11 $. Expand: $ 4(x+1)^2 - 4 - 9(y-1)^2 + 9 = 11 \Rightarrow 4(x+1)^2 - 9(y-1)^2 = 6 $. Rewrite: $ \frac{(x+1)^2}{\frac{3}{2}} - \frac{(y-1)^2}{\frac{2}{3}} = 1 $. Center at $ (-1, 1) $. \boxed{(-1, 1)}1. **A chemistry experiment requires 50 mL of a 10% salt solution. If you only have a 20% salt solution and distilled water, how much of each should be mixed to obtain the desired concentration?

📌 To create a 10% salt solution from a 20% salt solution, you can use the formula for mixing solutions. Let \( x \) be the volume of the 20% solution, and \( 50 - x \) be the volume of water.

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