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Home » \|\mathbfv_\perp\| = \sqrt{\|\mathbfv\|^2 - \left(rac - Crosslake

\|\mathbfv_\perp\| = \sqrt{\|\mathbfv\|^2 - \left(rac - Crosslake

📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 Solution: We use the identity \( p^3 + q^3 = (p + q)^3 - 3pq(p + q) \). First, compute \( pq \) using \( (p + q)^2 = p^2 + 2pq + q^2 \). Substituting, \( 10^2 = 58 + 2pq \) gives \( 100 = 58 + 2pq \), so \( pq = 21 \). Then, \( p^3 + q^3 = 10^3 - 3 \cdot 21 \cdot 10 = 1000 - 630 = 370 \). oxed{370}

📌 Question: Expand the product \( (2x - 3y)^2 \).

📌 Solution: Apply the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). Here, \( (2x)^2 - 2 \cdot 2x \cdot 3y + (3y)^2 = 4x^2 - 12xy + 9y^2 \). oxed{4x^2 - 12xy + 9y^2}

📌 Question: If \( x + rac{1}{x} = 5 \), what is the value of \( 2x^2 + rac{2}{x^2} \)?

📌 Solution: Square both sides: \( x^2 + 2 + rac{1}{x^2} = 25 \), so \( x^2 + rac{1}{x^2} = 23 \). Multiply by 2: \( 2x^2 + rac{2}{x^2} = 46 \). oxed{46}

📌 Question: Find the intersection of the lines \( y = 3x + 1 \) and \( y = -2x + 11 \).

📌 Solution: Set \( 3x + 1 = -2x + 11 \). Solve for \( x \): \( 5x = 10 \) â \( x = 2 \). Substitute into \( y = 3(2) + 1 = 7 \). The intersection is \( (2, 7) \). oxed{(2, 7)}

📌 Question: If \( m(m + n) = 24 \) and \( m + n = 8 \), what is the value of \( m \)?

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