Question

Explain why the inequality \(x^{2}-x+1<0\) has the empty set as the solution set.

Short answer

The inequality \(x^2 - x + 1 < 0\) has no solutions because the quadratic function is always positive.

Step-by-step solution

Analyze the quadratic function y = x² - x + 1

Consider the quadratic function given by the expression \(y = x^2 - x + 1\). This inequality requires us to determine where this function is less than zero.

Find the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). For our equation \(x^2 - x + 1 = 0\), we have \(a = 1\), \(b = -1\), and \(c = 1\). Therefore, the discriminant is: \[ D = (-1)^2 - 4(1)(1) = 1 - 4 = -3 \]

Interpret the discriminant

A negative discriminant (\(D < 0\)) means that the quadratic equation has no real roots, which implies that the quadratic function \(y = x^2 - x + 1\) does not intersect the x-axis. This indicates that the quadratic function is always either above or below the x-axis.

Determine the sign of the quadratic function

To determine whether the parabola opens upwards or downwards, look at the coefficient of the \(x^2\) term. Since the coefficient is positive (\(a = 1 > 0\)), the parabola opens upwards. Therefore, \(y = x^2 - x + 1\) is always greater than zero.

Conclusion

Since the quadratic function \(y = x^2 - x + 1\) is always greater than zero, there are no values of x for which \(x^2 - x + 1 < 0\). Thus, the solution set is empty.

Key concepts

discriminant

Understanding the concept of the discriminant is crucial for solving quadratic inequalities. The discriminant is a special value that can be calculated from the coefficients of a quadratic equation.In our case, the quadratic equation was given by y = x^{2}-x + 1. The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are the coefficients.The formula to find the discriminant (D) is: b^2 - 4ac.Let's substitute the coefficients from our equation: a = 1, b = -1, and c = 1. Plugging these values into the formula gives D = (-1)^2 - 4(1)(1) = 1 - 4 = -3.The discriminant helps us determine the nature of the roots (solutions) of the equation.In this case, D = -3, which is less than zero. A negative discriminant means that the quadratic equation has no real roots.This also suggests that the graph of the quadratic function does not intersect the x-axis.

quadratic function

A quadratic function is any function that can be written in the form ax^2 + bx + c, where a, b, and c are constants, and x is the variable.Quadratic functions form a parabola when graphed.To understand quadratic functions better, it's helpful to understand their graph.The direction in which the parabola opens is determined by the coefficient a. If a > 0, the parabola opens upwards.If a < 0, the parabola opens downwards.In the case of our function y = x^2 - x + 1, the coefficient a = 1 is positive.So, the parabola opens upwards.Given that we already determined that the discriminant is negative (indicating no real roots), this means our parabola does not touch or cross the x-axis.This tells us something about the inequality x^2 - x + 1 < 0 that we're trying to solve. Since the parabola always lies above the x-axis and opens upwards, our quadratic function y = x^2 - x + 1 is always greater than zero.

solution set

The solution set is where we find the values of x that satisfy the given inequality. For the inequality x^2 - x + 1 < 0, we need values where the quadratic function is less than zero. However, from the analysis of the quadratic function and its discriminant, we see that this particular function y = x^2 - x + 1 is always positive for all real numbers.That means there are no values of x for which the quadratic function becomes less than zero.So the inequality x^2 - x + 1 < 0 has no solutions in the real number system.Therefore, the solution set is empty.Understanding the solution set of quadratic inequalities is essential in identifying where a function's values lie relative to a given threshold—especially in advanced mathematical problems like optimization and modeling.