Multiple choice

Ramesh and Mahesh solve an equation. In solving Ramesh commits a mistake in constant term and find the roots are $8$ and $2$ Mahesh commits a mistake in the coefficient of x and find roots $-9$ and $-1$. The correct roots are

  1. $-8,2$
  2. $9,1$
  3. $9,-1$
  4. $-8,-2$
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B Correct answer
Explanation

Ramesh got roots 8 and 2, so the equation is (x-8)(x-2) = x^2 - 10x + 16. He made a mistake in the constant term, so the coefficient of x (-10) is correct. Mahesh got roots -9 and -1, so the equation is (x+9)(x+1) = x^2 + 10x + 9. He made a mistake in the coefficient of x, so the constant term (9) is correct. The correct equation is x^2 - 10x + 9 = 0. Factoring gives (x-9)(x-1) = 0, so roots are 9 and 1.

AI explanation

Ramesh made a mistake only in the constant term, meaning his coefficient of x is correct; therefore, the correct sum of the roots is the same as his calculated sum, which is 8 + 2 = 10. Mahesh made a mistake only in the coefficient of x, meaning his constant term is correct; therefore, the correct product of the roots is the same as his calculated product, which is (-9) multiplied by (-1), giving 9. We now know the correct sum of the roots is 10 and the correct product is 9. Setting up the quadratic equation with these values gives x^2 - 10x + 9 = 0. Factoring this equation yields (x - 9)(x - 1) = 0, resulting in the correct roots being 9 and 1.

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