What are the values of |x| - 2 = x^2?

To find the values of x that satisfy the equation |x| - 2 = x^2, we need to consider two cases for the absolute value part, because |x| can be x when x is non-negative, and -x when x is negative.

**Case 1: x is non-negative (x ≥ 0)** In this case, |x| = x. So the equation becomes: x - 2 = x^2 This simplifies to a quadratic equation: x^2 - x + 2 = 0

However, by looking at this equation, we see that the coefficient of the x^2 term and the constant term are both positive, which means the parabola opens upward and the vertex of the parabola will lie above the x-axis. Since there are no x-intercepts for this parabola, there are no real solutions to x^2 - x + 2 = 0.

**Case 2: x is negative (x < 0)** Here, |x| = -x. So the equation becomes: -x - 2 = x^2 This leads to the following quadratic equation: x^2 + x + 2 = 0

Similarly, the coefficients of x^2 and the constant term are both positive, indicating that the parabola corresponding to this equation will also open upward and have its vertex above the x-axis. This means that there are no real solutions to this equation as well.

To summarize, there are no real values of x that satisfy the equation |x| - 2 = x^2 because the resulting quadratic equations do not have real roots. It's also worth noting that you could formally prove this by applying the quadratic formula and showing that the discriminant (b^2 - 4ac) is negative in both cases, which indicates the absence of real roots.

Extra: The absolute value function, |x|, is defined as follows: |a| = { a if a ≥ 0, -a if a < 0 }

It retrieves the distance of a number a from zero on the number line without considering the direction. The absolute value is never negative.

In the context of solving equations involving absolute values, one has to consider all possibilities that can fulfill the equation due to the piecewise definition of the absolute value function. This typically involves splitting the problem into multiple cases.

A quadratic equation ax^2 + bx + c = 0, where a, b, and c are constants, can be solved by factoring, completing the square, or using the quadratic formula. The quadratic formula is given by: x = (-b ± √(b^2 - 4ac))/2a

The discriminant, b^2 - 4ac, helps determine the nature of the roots: - If the discriminant is positive, there are two distinct real roots. - If the discriminant is zero, there is exactly one real root. - If the discriminant is negative, there are no real roots (the roots are complex).