Quadratic Equations - Examples & Applications

This section provides step-by-step examples demonstrating how to solve quadratic equations using various analytical methods, including factoring, completing the square, and applying the quadratic formula.

Example 1: Solving by Factoring (Basic)

Problem: Solve the quadratic equation x2−8x+15=0x^2 - 8x + 15 = 0.

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Example 2: Factoring with a Leading Coefficient $a > 1$

Problem: Solve the quadratic equation 3x2+10x−8=03x^2 + 10x - 8 = 0.

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Example 3: Solving using the Quadratic Formula

Problem: Solve the quadratic equation 2x2−7x−4=02x^2 - 7x - 4 = 0.

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Example 4: Completing the Square

Problem: Solve the quadratic equation x2+6x−7=0x^2 + 6x - 7 = 0 by completing the square.

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Example 5: Utilizing the Discriminant

Problem: Determine the number and nature of the solutions for the equation 2x2−5x+3=02x^2 - 5x + 3 = 0 without solving it.

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Example 6: Applying Vieta's Formulas

Problem: Without solving the equation, find the sum and product of the roots for 2x2−5x−3=02x^2 - 5x - 3 = 0.

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