Question

При якому значенні пааметра а рівняння (|х|-2)^2-1=а має три корені?

Answer 1

Equation with Three Roots

To find the value of parameter a for which the equation (|x|-2)^2-1=a has three roots, we can analyze the equation and its properties.

The equation (|x|-2)^2-1=a is a quadratic equation in terms of x. To determine the number of roots, we can examine the discriminant of the quadratic equation. The discriminant is given by b^2-4ac, where a, b, and c are the coefficients of the quadratic equation.

In this case, the equation can be rewritten as x^2-4|x|+3-a=0. Comparing this equation with the standard form ax^2+bx+c=0, we have a=1, b=-4, and c=3-a.

The discriminant is then calculated as (-4)^2-4(1)(3-a), which simplifies to 16-12+4a or 4a+4.

To have three roots, the discriminant must be greater than zero. Therefore, we need 4a+4>0.

Solving this inequality, we have:

4a+4>0

Subtracting 4 from both sides:

4a>-4

Dividing both sides by 4:

a>-1

Therefore, for the equation (|x|-2)^2-1=a to have three roots, the parameter a must be greater than -1.