Quadratic Equations

x = 3, x = 1/2

Step 1: For 2x² - 7x + 3 = 0, a = 2, b = -7, c = 3. Step 2: Calculate discriminant D = (-7)² - 4(2)(3) = 49 - 24 = 25. Step 3: Apply quadratic formula x = (7 ± √25)/(2×2) = (7 ± 5)/4. Step 4: So x = (7 + 5)/4 = 12/4 = 3 or x = (7 - 5)/4 = 2/4 = 1/2. Step 5: The solutions are x = 3 and x = 1/2.

x² - 4x + 3 = 0

Step 1: For real distinct roots, discriminant D > 0. Step 2: Check first equation: D = (-4)² - 4(1)(3) = 16 - 12 = 4 > 0. Step 3: Check second: D = (-4)² - 4(1)(4) = 0 (equal roots). Step 4: Check third: D = 2² - 4(1)(5) = -16 < 0 (no real roots). Step 5: Check fourth: D = (-4)² - 4(2)(2) = 0 (equal roots). Only the first has D > 0.

x = 3, x = -2

Step 1: For the equation (x - 3)(x + 2) = 0, use the zero product property. Step 2: Set each factor equal to zero: x - 3 = 0 or x + 2 = 0. Step 3: Solve first equation: x = 3. Step 4: Solve second equation: x = -2. Step 5: Therefore, the roots are x = 3 and x = -2.

3x² - 2x - 1 = 0

Step 1: Standard form of quadratic equation is ax² + bx + c = 0. Step 2: Start with 3x² = 2x + 1. Step 3: Move all terms to left side: 3x² - 2x - 1 = 0. Step 4: Check: a = 3, b = -2, c = -1, which gives the standard form. Step 5: The equation 3x² - 2x - 1 = 0 is in standard form.