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The roots of a quadratic equation are -2/3 and -1/4. Find the equation.

Mathematics

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Answers

2 parts
1

B\text{B}

2

C\text{C}

Step-by-step solution

9 steps

Here are the solutions for questions 1 and 2.

Part 1· 5 steps

  1. Identify the given roots of the quadratic equation.

    The roots are x1=−23x_1 = -\frac{2}{3} and x2=−14x_2 = -\frac{1}{4}.

  2. Calculate the sum of the roots.

    Sum=x1+x2=−23+(−14)=−23−14\text{Sum} = x_1 + x_2 = -\frac{2}{3} + \left(-\frac{1}{4}\right) = -\frac{2}{3} - \frac{1}{4}

    To add these fractions, find a common denominator, which is 12.

    Sum=−2×43×4−1×34×3=−812−312=−1112\text{Sum} = -\frac{2 \times 4}{3 \times 4} - \frac{1 \times 3}{4 \times 3} = -\frac{8}{12} - \frac{3}{12} = -\frac{11}{12}

  3. Calculate the product of the roots.

    Product=x1×x2=(−23)×(−14)=2×13×4=212=16\text{Product} = x_1 \times x_2 = \left(-\frac{2}{3}\right) \times \left(-\frac{1}{4}\right) = \frac{2 \times 1}{3 \times 4} = \frac{2}{12} = \frac{1}{6}

  4. Form the quadratic equation using the formula x2−(Sum)x+(Product)=0x^2 - (\text{Sum})x + (\text{Product}) = 0.

    x2−(−1112)x+16=0x^2 - \left(-\frac{11}{12}\right)x + \frac{1}{6} = 0 x2+1112x+16=0x^2 + \frac{11}{12}x + \frac{1}{6} = 0

  5. Clear the denominators by multiplying the entire equation by the least common multiple (LCM) of 12 and 6, which is 12.

    12(x2+1112x+16)=12×012 \left(x^2 + \frac{11}{12}x + \frac{1}{6}\right) = 12 \times 0 12x2+12(1112)x+12(16)=012x^2 + 12 \left(\frac{11}{12}\right)x + 12 \left(\frac{1}{6}\right) = 0 12x2+11x+2=012x^2 + 11x + 2 = 0

    Comparing this with the given options, the correct equation is B.
    The final answer is B\boxed{\text{B}}.

Part 2

C\text{C}

  1. Calculate the initial total sum of ages.

    Number of learners = 14
    Sum of ages = 168

  2. Determine the new number of learners and the new mean age.

    A new learner is added, so the new number of learners = 14+1=1514 + 1 = 15.
    The new mean age = 12.2.

  3. Calculate the new total sum of ages.

    The formula for mean is Mean=Sum of agesNumber of learners\text{Mean} = \frac{\text{Sum of ages}}{\text{Number of learners}}.
    So, New Sum of ages=New Mean×New Number of learners\text{New Sum of ages} = \text{New Mean} \times \text{New Number of learners}

    New Sum of ages=12.2×15\text{New Sum of ages} = 12.2 \times 15 New Sum of ages=183\text{New Sum of ages} = 183

  4. Calculate the age of the new learner.

    The age of the new learner is the difference between the new total sum of ages and the initial total sum of ages.

    Age of new learner=New Sum of ages−Initial Sum of ages\text{Age of new learner} = \text{New Sum of ages} - \text{Initial Sum of ages} Age of new learner=183−168\text{Age of new learner} = 183 - 168 Age of new learner=15\text{Age of new learner} = 15

    Comparing this with the given options, the correct age is C.
    The final answer is C\boxed{\text{C}}.

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