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Find the quadratic equation with roots 2/3 and -1, and choose the correct option.

Mathematics

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Step-by-step solution

5 steps

  1. Identify the given roots.

    Let the roots be α=23\alpha = \frac{2}{3} and β=−1\beta = -1.

  2. Calculate the sum of the roots.

    α+β=23+(−1)=23−33=−13\alpha + \beta = \frac{2}{3} + (-1) = \frac{2}{3} - \frac{3}{3} = -\frac{1}{3}

  3. Calculate the product of the roots.

    αβ=23×(−1)=−23\alpha \beta = \frac{2}{3} \times (-1) = -\frac{2}{3}

  4. Form the quadratic equation using the formula x2−(α+β)x+αβ=0x^2 - (\alpha + \beta)x + \alpha\beta = 0.

    Substitute the sum and product of the roots into the formula:

    x2−(−13)x+(−23)=0x^2 - \left(-\frac{1}{3}\right)x + \left(-\frac{2}{3}\right) = 0 x2+13x−23=0x^2 + \frac{1}{3}x - \frac{2}{3} = 0

  5. Multiply the entire equation by 3 to eliminate the fractions.

    3(x2+13x−23)=3×03 \left(x^2 + \frac{1}{3}x - \frac{2}{3}\right) = 3 \times 0 3x2+x−2=03x^2 + x - 2 = 0

    Comparing this with the given options:
    A. 3x2−x−2=03x^2 - x - 2 = 0
    B. 3x2+x+2=03x^2 + x + 2 = 0
    C. 3x2+x−2=03x^2 + x - 2 = 0
    D. 3x2+x−1=03x^2 + x - 1 = 0

    The derived equation matches option C.

    The final answer is C. 3x2+x−2=0\boxed{\text{C. } 3x^2 + x - 2 = 0}.

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