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Algebra

2x² + 5x - 3 = 0

Step-by-Step Explanation

1. 1. Split the middle term 5x into 6x - x: 2x² + 6x - x - 3 = 0
2. 2. Factor by grouping terms: 2x(x + 3) - 1(x + 3) = 0
3. 3. Combine the factors to complete factorization: (2x - 1)(x + 3) = 0
4. 4. Solve for x when each factor is set to 0: x = 1/2 or x = -3

Calculus

∫ x² dx

Step-by-Step Explanation

1. 1. Apply the power rule for integration: ∫ xⁿ dx = [xⁿ⁺¹ / (n + 1)] + C
2. 2. Here n = 2, so increase the power by 1 and divide by the new power: x²⁺¹ / (2 + 1)
3. 3. Compute to get the final indefinite integral: x³ / 3 + C (C is the constant of integration)

Geometry

Distance between (1,2) and (4,6)

Step-by-Step Explanation

1. 1. Apply the Euclidean distance formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²]
2. 2. Substitute the coordinate values: d = √[(4 - 1)² + (6 - 2)²]
3. 3. Simplify the terms inside the square root: d = √[3² + 4²] = √[9 + 16] = √25
4. 4. Find the square root to get the distance: d = 5

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