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Chapter 2: Problem 38

Solve the inequality algebraically or graphically. $$x^{2}-x+1 \geq 0$$

Short Answer

Expert verified
The inequality \(x^{2}-x+1 \geq 0\) holds for all real numbers.

Step by step solution

01

Find Roots

Set \(x^{2}-x+1 = 0\) and solve for \(x\). This will yield complex roots as the discriminant \(b^2 - 4ac = (-1)^2 - 4(1)(1) = -3\) is negative. Hence, there are no real roots.
02

Screening Intervals

In case of no real roots, the quadratic function does not intersect with the x-axis. Due to the positive coefficient of the \(x^{2}\) term, the parabola opens upwards, implying that the function is always greater than or equal to zero.
03

Conclusion

Given that the function is always greater than or equal to zero, for all real numbers \(x\), it follows that \(x^{2}-x+1 \geq 0\). So the solution set is the set of all real numbers.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Roots
Many students encounter the term "complex roots" in the context of solving quadratic equations. It's essential to understand what it means. Quadratic equations can be solved using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Roots are determined by the expression under the square root sign, called the discriminant. When the discriminant is negative, the square root of a negative number arises, leading to complex numbers, which are not real.
Complex numbers are expressed in the form of \(a + bi\), where \(i\) is the imaginary unit, defined as the square root of -1.
Effectively, if a quadratic has complex roots, it does not intersect the x-axis.
Discriminant
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\).
The value of the discriminant tells us about the nature of the roots of the quadratic equation.
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
  • If it is exactly zero, the equation has one real root (a repeated root).
  • If it is negative, as in the example given, the equation has complex roots.
A negative discriminant means that the graph of the quadratic function doesn't cross the x-axis.
That's a quick signal the function's values are either entirely above or entirely below the x-axis.
Parabola Opens Upwards
A quadratic function, represented as \(ax^2 + bx + c\), can form a parabolic graph.
The direction in which the parabola opens is determined by the value of \(a\).
  • If \(a > 0\), the parabola opens upwards like a cup.
  • If \(a < 0\), it opens downwards, resembling an upside-down bowl.
In our example, the coefficient \(a = 1\), meaning the parabola opens upwards. An upwards-opening parabola ensures the vertex is at the lowest point on the graph, suggesting all function values are above or at this point.
Since the parabola does not dip below the x-axis and opens upwards, it's always \(\geq 0\) for any real number \(x\).
This is why the solution to the inequality \(x^2 - x + 1 \geq 0\) includes all real numbers, confirming the function remains nonnegative throughout.

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