Multiple choice

If the sum and product of the roots of the quadratic equation $ax^2-5x+c=0$ are both equal to $10$, then find the values of a and c.

  1. $a=\dfrac 12,c=5$
  2. $a=\dfrac 12,c=2$
  3. $a=\dfrac 12,c=8$

  4. None of these
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A Correct answer
Explanation

For ax^2 - 5x + c = 0, sum of roots = 5/a = 10 => a = 0.5. Product of roots = c/a = 10 => c/0.5 = 10 => c = 5.

AI explanation

For a standard quadratic equation ax squared minus 5x plus c equals 0, Vieta's formulas state that the sum of the roots equals 5 over a and the product of the roots equals c over a. We are given that both the sum and product are equal to 10, leading to the two equations 5 over a equals 10 and c over a equals 10. Solving the first equation gives a equal to one half, and substituting this into the second equation yields c equal to 5.

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