Quadratic Equations

The equation has two equal real roots

Step 1: When discriminant Δ = b² - 4ac = 0, the quadratic formula becomes x = (-b ± √0)/2a = (-b ± 0)/2a. Step 2: This simplifies to x = -b/2a for both values. Step 3: Since both roots are identical, we say the equation has two equal real roots or a repeated root. Step 4: This occurs when the parabola just touches the x-axis at one point. Step 5: Example: x² - 6x + 9 = 0 has discriminant 36 - 36 = 0 and roots x = 3, x = 3.

x = -2, x = -4

Step 1: Need to factor x² + 6x + 8. Step 2: Find two numbers that multiply to 8 and add to 6. These are 2 and 4. Step 3: Rewrite: x² + 6x + 8 = x² + 2x + 4x + 8 = x(x + 2) + 4(x + 2) = (x + 2)(x + 4). Step 4: Set each factor to zero: x + 2 = 0 gives x = -2, and x + 4 = 0 gives x = -4. Step 5: Verify: (-2)² + 6(-2) + 8 = 4 - 12 + 8 = 0 ✓ and (-4)² + 6(-4) + 8 = 16 - 24 + 8 = 0 ✓

7/3

Step 1: For a quadratic equation ax² + bx + c = 0, the sum of roots = -b/a. Step 2: From 3x² - 7x + 2 = 0, we have a = 3, b = -7, c = 2. Step 3: Sum of roots = -(-7)/3 = 7/3. Step 4: This relationship comes from expanding (x - α)(x - β) = x² - (α + β)x + αβ, where α and β are the roots. Step 5: We can verify by finding actual roots: using quadratic formula, roots are 2 and 1/3, and 2 + 1/3 = 7/3 ✓

6

Step 1: For a quadratic equation ax² + bx + c = 0, the product of roots = c/a. Step 2: From x² - 5x + 6 = 0, we have a = 1, b = -5, c = 6. Step 3: Product of roots = c/a = 6/1 = 6. Step 4: This relationship comes from the fact that when we expand (x - α)(x - β), the constant term equals αβ. Step 5: We can verify: the roots are 2 and 3, and 2 × 3 = 6 ✓