Construct a quadratic equation whose roots are 1over3 and -1

Mathematics

ScanSolve AI Answer

Answer

3x2+2x−1=03x^2 + 2x - 1 = 0

Step-by-step solution

5 steps

  1. Identify the roots.

    Let the roots be α=13\alpha = \frac{1}{3} and β=−1\beta = -1.

  2. Calculate the sum of the roots.

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

  3. Calculate the product of the roots.

    αβ=(13)×(−1)=−13\alpha\beta = \left(\frac{1}{3}\right) \times (-1) = -\frac{1}{3}

  4. Use the formula for a quadratic equation given its roots

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

  5. Clear the fractions by multiplying the entire equation by the least common multiple of the denominators, which is 3.

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