Multiple choice

If $ x= \dfrac{2}{3} $ and $x=-3$ are the roots of the equation $ax^2+7x +b=0$, find the values of $a$ and $b.$

  1. $a=3,b=-6$
  2. $a=21, b=42$
  3. $a=2, b=1$
  4. $a=1, b=2$
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A Correct answer
Explanation

For the equation ax^2 + 7x + b = 0, the sum of roots is -7/a and the product is b/a. Roots are 2/3 and -3. Sum: 2/3 - 3 = -7/3 = -7/a, so a = 3. Product: (2/3) * (-3) = -2 = b/a = b/3, so b = -6.

AI explanation

For the quadratic equation ax^2 + 7x + b = 0, the sum and product of the roots 2/3 and -3 are (2/3) + (-3) = -7/3 and (2/3)(-3) = -2 respectively. Using Vieta's formulas, -7/a = -7/3 which gives a = 3, and b/a = -2 which gives b = -6. Thus, the values are a = 3 and b = -6.

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