Multiple choice

If the sum and product of roots of the quadratic equation $\displaystyle { ax }^{ 2 }-5x+c$ are both equal to 10, find the value of $a$ and $c$.

  1. $\displaystyle 2,3$
  2. $\displaystyle \frac { 1 }{ 3 } ,5$
  3. $\displaystyle \frac { 1 }{ 2 } ,5$
  4. $\displaystyle 3,5$
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C Correct answer
Explanation

For ax^2 - 5x + c = 0, the sum of roots is 5/a = 10, so a = 1/2. The product of roots is c/a = 10, so c = 10 * (1/2) = 5.

AI explanation

Using the relationship between roots and coefficients for the quadratic equation ax^2 - 5x + c = 0, the sum of the roots is 5/a and the product is c/a. We are given that both the sum and product equal 10, so we set up the equations 5/a = 10 and c/a = 10. Solving the first equation yields a = 1/2, and substituting this into the second equation yields c = 5. The values are a = 1/2 and c = 5.

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