Question

При каком значении параметра a уравнение |x^2-2x-3|=a имеет 3 корня?

Answer 1

Задание № 2:

При каком значении параметра a уравнение |x^2−2x−3|=a имеет три корня?

введем функцию

y=|x^2−2x−3|

рассмотрим функцию без модуля

y=x^2−2x−3

y=(x−3)(х+1)

при х=3 и х=-1 - у=0

х вершины = 2/2=1

у  вершины = 1-2-3=-4

после применения модуля график отражается в верхнюю полуплоскость

при а=0 - 2 корня (нули х=3 и х=-1)

при 0<а<4 - 4 корня (2 от исходной параболы, 2 от отображенной части)

при а=4 - 3 корня (2 от исходной параболы, 1 от вершины х=1)

при а>4 - 2 корня (от исходной параболы)

ответ: 4

Answer 2

У меня было в ответе 4

Answer 3

Finding the Number of Roots for the Equation |x^2-2x-3|=a

To find the number of roots for the equation |x^2-2x-3|=a, we can use the discriminant of the quadratic equation. The discriminant is given by the expression b^2-4ac, where a, b, and c are the coefficients of the quadratic equation ax^2+bx+c=0.

The given equation |x^2-2x-3|=a can be rewritten as x^2-2x-3=a or x^2-2x-3-a=0. Comparing this with the standard form of a quadratic equation ax^2+bx+c=0, we have a=1, b=-2, and c=(-3-a).

The discriminant D of the quadratic equation ax^2+bx+c=0 is given by D = b^2-4ac.

Let's calculate the discriminant D and determine the number of roots based on its value.

The discriminant D is calculated as follows: D = (-2)^2 - 4(1)(-3-a) D = 4 + 12 + 4a D = 16 + 4a

Now, we can determine the number of roots based on the value of the discriminant D.

- If D > 0, the equation has two distinct real roots. - If D = 0, the equation has exactly one real root. - If D < 0, the equation has no real roots.

Conclusion

Based on the value of the discriminant D = 16 + 4a, we can determine the number of roots for the equation |x^2-2x-3|=a. If D > 0, the equation has two distinct real roots; if D = 0, the equation has exactly one real root; and if D < 0, the equation has no real roots.

Note: The specific value of parameter 'a' will determine the number of roots for the given equation.