Find the quadratic equation whose roots are $\frac{2}{3}$ and - 1

Analyze the question: We need a quadratic equation whose roots (solutions) are 2/3 and -1.

Concept: If the roots are α and β, the equation is

x² - (α + β)x + αβ = 0

Solution:

Sum of roots: 2/3 + (-1) = 2/3 - 1 = -1/3

Product of roots: (2/3)(-1) = -2/3

Substitute into the formula:

x² - (-1/3)x + (-2/3) = 0 x² + (1/3)x - 2/3 = 0

Multiply through by 3 to clear the fractions:

3x² + x - 2 = 0

Check by factorising: 3x² + x - 2 = (3x - 2)(x + 1)

• 3x - 2 = 0 gives x = 2/3 ✓
• x + 1 = 0 gives x = -1 ✓

Tip: Always use "x² - (sum)x + product = 0". Be careful with the sign in front of the sum. A fast check is to substitute one root, such as x = -1, into the options: for C, 3(1) + (-1) - 2 = 0 ✓.