If the mean and the standard deviation of the data 3, 5, 7, a, b are 5 and 2 respectively, then a and b are the roots of the equation:
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x2 - 20x + 18 = 0
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2x2 - 20x + 19 = 0
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x2 - 10x + 18 = 0
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x2 - 10x + 19 = 0
Mean = (3+5+7+a+b)/5 = 5 => 15+a+b = 25 => a+b = 10. Variance = 2^2 = 4. Variance = [ (3-5)^2 + (5-5)^2 + (7-5)^2 + (a-5)^2 + (b-5)^2 ] / 5 = 4. 4 + 0 + 4 + (a-5)^2 + (b-5)^2 = 20 => (a-5)^2 + (b-5)^2 = 12. Using (a-5)^2 + (b-5)^2 = a^2+b^2 - 10(a+b) + 50 = 12. Since (a+b)^2 = a^2+b^2 + 2ab = 100, a^2+b^2 = 100-2ab. Substitute: 100-2ab - 100 + 50 = 12 => 2ab = 38 => ab = 19. Equation is x^2 - (a+b)x + ab = 0 => x^2 - 10x + 19 = 0.
Since the mean of the 5 data points is 5, their sum is 25; therefore, 3 + 5 + 7 + a + b = 25, which simplifies to a + b = 10. The variance formula gives the standard deviation squared as the average of the squared deviations, so the sum of squared deviations is 5 multiplied by 4 (since the variance is 2 squared) equals 20. Expanding the squared deviations, (3-5)^2 + (5-5)^2 + (7-5)^2 + (a-5)^2 + (b-5)^2 = 20, which results in 8 + (a-5)^2 + (b-5)^2 = 20, meaning (a-5)^2 + (b-5)^2 = 12. Substituting b = 10 - a into this equation yields a quadratic equation whose roots are a and b, specifically x^2 - 10x + 19 = 0.