Determine the values of the variable for which the expression is defined as a real number. (Enter your answer using interval notation.)
x2 − 5x − 50
To determine the values of x for which the expression is defined as a real number, we need to find the values of x that make the discriminant of the quadratic equation \(x^2 -5x - 50\) nonnegative.
The discriminant is given by \(b^2 - 4ac\), where \(a = 1\), \(b = -5\), and \(c = -50\).
Substituting these values into the discriminant formula, we have \((-5)^2 - 4(1)(-50) = 25 + 200 = 225\).
Since the discriminant is positive, the quadratic equation has two distinct real roots. Therefore, the expression \(x^2 - 5x - 50\) is defined as a real number for all values of x.
Using interval notation, the values of x for which the expression is defined are \((-∞, ∞)\).
To determine the values of the variable, x, for which the expression x^2 -5x - 50 is defined as a real number, we need to find the interval(s) where the expression does not result in complex numbers.
To do that, we can check the discriminant of the quadratic equation. The discriminant, denoted as Δ, is given by the formula: Δ = b^2 - 4ac
The quadratic equation is in the form of ax^2 + bx + c, where:
- a = 1
- b = -5
- c = -50
Let's calculate the discriminant: Δ = (-5)^2 - 4(1)(-50)
= 25 + 200
= 225
Since the discriminant is positive (225 > 0), it means that the quadratic equation has two distinct real roots. Therefore, the expression x^2 -5x - 50 is defined for all real numbers.
In interval notation, we can express the solution as: (-∞, ∞)