Quadratic Equations and Linear Inequalities

5/3

Step 1: For ax² + bx + c = 0, sum of roots α+β = -b/a. Step 2: Here a=3, b=-5, c=9. Step 3: Sum of roots = -(-5)/3 = 5/3. Step 4: Note the formula is -b/a, so the negative of the coefficient of x divided by the coefficient of x². Step 5: Therefore α+β = 5/3.

3

Step 1: For ax² + bx + c = 0, product of roots αβ = c/a. Step 2: Here a=3, b=-5, c=9. Step 3: Product of roots = c/a = 9/3 = 3. Step 4: This is the constant term divided by the leading coefficient. Step 5: Therefore αβ = 3. Do not confuse with sum of roots which is -b/a.

x² - 4x + 3 = 0

Step 1: A quadratic equation with roots α and β is written as x² - (α+β)x + αβ = 0. Step 2: Substitute the given values: α+β = 4 and αβ = 3. Step 3: The equation becomes x² - 4x + 3 = 0. Step 4: Verify: roots of x²-4x+3=0 are x=1 and x=3. Sum=1+3=4 ✓, Product=1×3=3 ✓. Step 5: The sign of the sum term is negative (−(α+β)), which is a common mistake to watch.

-29/9

Step 1: α+β = 5/3 and αβ = 3 (from the equation 3x²-5x+9=0). Step 2: Use the identity α²+β² = (α+β)² - 2αβ. Step 3: (α+β)² = (5/3)² = 25/9. Step 4: 2αβ = 2×3 = 6 = 54/9. Step 5: α²+β² = 25/9 - 54/9 = -29/9. The negative value is correct and is possible because α and β are complex numbers here (D<0).