Question

Find the solution set of \(-x^{2}-4 x-5>0\)

Short answer

The solution set for the inequality \(-x^2 - 4x - 5 > 0\) is the empty set, as there are no values of x that satisfy the inequality.

Step-by-step solution

Solve the related equation

To find the critical points, we need to solve the equation: \[-x^2 - 4x - 5 = 0\]. Since this is a quadratic equation, we can use the Quadratic Formula: \[x = \frac {-b\pm\sqrt{b^{2}-4ac}}{2a}\]. In this case, \(a = -1\), \(b = -4\), and \(c = -5\).

Calculate the discriminant

Before applying the quadratic formula, we'll calculate the discriminant, which is given by the part inside the square root: \[D = b^2 - 4ac\]. Plugging in \(a = -1\), \(b = -4\), and \(c = -5\), we find: \[D = (-4)^2 - 4(-1)(-5) = 16 - 20 = -4\]

Analyze the discriminant

The discriminant is negative, which means that the related quadratic equation has no real roots.¶

Determine the solution set

Since the related quadratic equation has no real roots, the inequality either holds true for all x or does not hold true for any x. To determine which, we'll pick a test point and see if it satisfies the inequality. Let's pick \(x = 0\): \[- (0)^2 - 4(0) - 5 = -5\]. Since \(-5 > 0\) is false, the inequality does not hold true for any x. Therefore, the solution set is empty, and there are no values of x for which the inequality holds true. In conclusion, the solution set for \(-x^2 - 4x - 5 > 0\) is the empty set, as there are no values of x that satisfy the inequality.

Key concepts

Discriminant Analysis

Understanding how to solve quadratic inequalities often involves the concept of discriminant analysis, which is a critical step in determining the nature of the roots of a quadratic equation. The discriminant is a component of the quadratic formula and is represented as 'D' in the equation \(D = b^2 - 4ac\). What this means is pretty straightforward: it's the part under the square root in the quadratic formula.

The value of the discriminant gives us crucial information about the roots. If the discriminant is positive (\(D > 0\)), there are two distinct real roots. If it's zero (\(D = 0\)), there's exactly one real root (which is repeated). However, when the discriminant is negative (\(D < 0\)), as in our exercise with \(D = -4\), this indicates that there are no real roots to the equation.

In terms of inequalities and determining the solution sets, the discriminant tells us whether there are critical points on the graph where the parabola intersects the x-axis. No real roots mean the parabola does not cross the x-axis, which in turn affects the range of x values satisfying the inequality.

Quadratic Formula

The quadratic formula is a fundamental tool used to solve quadratic equations of the form \(ax^2 + bx + c = 0\). It is given by the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). With the quadratic formula, solving any quadratic equation becomes a systematic process.

In our example, the coefficients are \(a = -1\), \(b = -4\), and \(c = -5\). By plugging these into the quadratic formula, we attempt to find the values of x that make the equation true. However, with a negative discriminant, this particular equation has no solution in the real number system.

The importance of this formula doesn't diminish even when we don't find real roots; it's still integral to understanding the structure of the quadratic expression and provides a pathway to resolution when dealing with inequalities.

Solution Sets

Solution sets are the collection of all possible solutions that satisfy a given inequality. In the case of quadratic inequalities, such as \( -x^2 - 4x - 5 > 0 \), the solution set can be thought of as the range of x-values for which the inequality holds true. If we were dealing with a quadratic equation (equal to zero), finding the roots would be our goal, but with an inequality, we are interested in where the quadratic expression is either above or below the x-axis.

When there are real roots, the solution set is usually an interval or a pair of intervals where the inequality is satisfied. However, if the discriminant reveals no real roots, as in our example, we're essentially determining whether the entire parabola is above or below the x-axis. Because we determined that there are no x-values that satisfy the inequality, the solution set is empty, illustrating that a proper analysis of discriminant and quadratic formula use is crucial for determining correct solution sets.