Kent and Agnes try to solve a quadratic equation ax^2+bx+c=0,where a,b and c are integers. Kent copies the value of b wrongly. The roots that he gets are 2 and -5/3. Agnes copies the value of c wrongly. The roots that she gets are -3 and 2/3. Find the original equation.
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Verified answer
For Kent, he thinks the equation is
(x - 2)(x + 5/3) = 0
(x - 2)(3x + 5)/3 = 0
(x - 2)(3x + 5) = 0
3x^2 - x - 10 = 0
so a = 3 and c = - 10 (only b is wrong).
For Agnes, she thinks the equation is
(x + 3)(x - 2/3) = 0
(x + 3)(3x - 2)/3 = 0
(x + 3)(3x - 2) = 0
3x^2 + 7x - 6 = 0
so a = 3 and b = 7 (only c is wrong).
So the original equation is 3x^2 + 7x - 10 = 0
For Kent, as b is wrong, using product of roots, we have,
2 * (-5/3)＝c/a
==> a：c＝3：-10
For Agnes, as c is wrong, using sum of roots, we have,
-3＋2/3＝-b/a
==> a：b＝3：7
so, a：b：c＝3：7：-10
Therefore, the original equation is ：
3x^2＋7x－10＝0