Problem 109 - A Graphical Approach to College Algebra

Question

Solve each inequality analytically. Support your answers graphically. Give exact values for endpoints. (a) \(x^{2}-x+1<0\) (b) \(x^{2}-x+1 \geq 0\)

Step-by-Step Solution

Answer

(a) No real solution; (b) True for all real \(x\).

Step 1: Understand the Inequalities

We are given two inequalities to solve: (a) \(x^2-x+1<0\) and (b) \(x^2-x+1\geq0\). Our task is to find the values of \(x\) that satisfy each of these inequalities.

Step 2: Identify the Nature of the Quadratic Expression

The expression \(x^2-x+1\) is a quadratic expression. To understand the intervals where it is positive or negative, examine its graph or calculate its discriminant \(b^2 - 4ac\), where \(a=1\), \(b=-1\), and \(c=1\). Here, the discriminant is \((-1)^2 - 4(1)(1)\), which equals \(1 - 4 = -3\). A negative discriminant means the quadratic has no real roots and it does not intersect the x-axis.

Step 3: Analyze (a): Solve \(x^2-x+1<0\)

Since the quadratic \(x^2-x+1\) does not intersect the x-axis (as we found the discriminant to be negative), it does not change sign. Analyze the coefficient of \(x^2\), which is positive. Hence, the quadratic is always positive for all real \(x\). Therefore, \(x^2-x+1<0\) has no solution since the expression is never negative.

Step 4: Analyze (b): Solve \(x^2-x+1 \geq 0\)

From the previous step, we know that the expression \(x^2-x+1\) is always positive as it does not intersect the x-axis and opens upwards. Therefore, \(x^2-x+1 \geq 0\) is true for all real \(x\).

Step 5: Graphical Representation

Graph the quadratic function \(y = x^2-x+1\). The parabola opens upwards and stays above the x-axis confirming our analytical solution: it never becomes negative. This visually shows that \(x^2-x+1 < 0\) has no solutions and \(x^2-x+1 \geq 0\) holds for all \(x\).

Key Concepts

Discriminant

The discriminant plays a crucial role in solving quadratic inequalities. It is derived from the formula \(b^2 - 4ac\), where \(a, b,\) and \(c\) are coefficients of the quadratic equation \(ax^2 + bx + c\). In our exercise, the quadratic expression \(x^2-x+1\) has a discriminant of \((-1)^2 - 4(1)(1)\), which simplifies to \(1 - 4 = -3\).

Here, the discriminant is negative. A negative discriminant indicates that the quadratic equation has no real roots. This is because the square root of a negative number is not defined in the realm of real numbers. Consequently, this suggests that the parabola represented by the quadratic does not touch or intersect the x-axis. Hence, the nature of the roots directly impacts whether the quadratic expression can change its sign. In this case, it stays consistently positive or negative depending on the leading coefficient. Understanding this concept helps in determining the solution to quadratic inequalities.

Graphical Solutions

Graphical solutions provide a visual method to understand quadratic inequalities. They involve plotting the graph of the relevant quadratic function. For the quadratic \(y = x^2 - x + 1\), the plot is a parabola.

Since the discriminant was negative, this parabola does not intersect the x-axis, indicating no real roots. Because the coefficient of \(x^2\) is positive, the parabola opens upwards, meaning it remains above the x-axis for all values of \(x\). This tells us that the expression is never negative.

• The graph for \(x^2-x+1<0\) shows no intersection below the x-axis, confirming no solutions exist for this inequality.
• Conversely, for \(x^2-x+1 \geq 0\), the graph confirms that all real \(x\) are solutions as the curve never dips below the x-axis.

Graphical solutions enable us to cross-verify analytical solutions visually, making abstraction more concrete.

Quadratic Functions

Quadratic functions are polynomial functions of degree 2, expressed in the standard form \(ax^2 + bx + c\). Our quadratic \(x^2 - x + 1\) exemplifies this.

Key characteristics of quadratic functions include their graphical representation as parabolas, which can open upwards or downwards:

• If the leading coefficient \(a\) is positive, the parabola opens upwards.
• If \(a\) is negative, it opens downwards.

For quadratic inequalities, understanding whether the parabola opens upwards or downwards helps determine intervals where the function is positive or negative. In this exercise, analyzing the sign and nature of the quadratic helped in solving the inequalities accurately.