Question

In Problems \(7-22,\) solve each inequality. 17\. \(x^{2}-x+1 \leq 0\)

Short answer

There is no solution to the inequality \(x^{2}-x+1 \leq 0\).

Step-by-step solution

Identify the inequality

The given inequality is: \(x^{2}-x+1 \leq 0\)

Analyze the quadratic expression

The quadratic expression to analyze is: \(x^{2}-x+1\)

Calculate the discriminant

The discriminant of a quadratic expression \(ax^{2}+bx+c\) is given by: \(D = b^{2}-4ac\). For \(x^{2}-x+1\), we have:\[ a = 1, b = -1, c = 1 \] Therefore, \(D = (-1)^{2} - 4 \times 1 \times 1 = 1 - 4 = -3\). Since the discriminant is negative, the quadratic expression has no real roots.

Determine the sign of the quadratic expression

Since the discriminant \(D\) is negative, the quadratic expression \(x^{2}-x+1\) does not change signs and is always positive for all real values of \(x\).

Conclude the inequality solution

Given that \(x^{2}-x+1 > 0\) for all real values of \(x\), there is no value of \(x\) that satisfies \(x^{2}-x+1 \leq 0\). Hence, there is no solution to the inequality.

Key concepts

quadratic equations

A quadratic equation takes the form of \( ax^2 + bx + c = 0 \). Let's break that down:

• \( a \): coefficient of \( x^2 \)
• \( b \): coefficient of \( x \)
• \( c \): constant term

The curve representing a quadratic equation is a parabola. The direction of the parabola (upward or downward) depends on the coefficient \(a\):

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

Solving a quadratic equation means finding the values of \( x \) (roots) that make the equation true. This comes handy when we deal with inequalities involving quadratics.

discriminant

The discriminant, denoted as \( D \), helps us determine the nature of the roots of a quadratic equation. The formula for the discriminant is given by: \( D = b^2 - 4ac \). Let's see what the value of \( D \) can tell us:

• If \( D > 0 \), the quadratic equation has two distinct real roots.
• If \( D = 0 \), the equation has exactly one real root (also known as a repeated or double root).
• If \( D < 0 \), the equation has no real roots; it has two complex (imaginary) roots.

In the original exercise, the discriminant for the quadratic expression \( x^2 - x + 1 \) is calculated as follows: \[D = (-1)^2 - 4 \times 1 \times 1 = 1 - 4 = -3\]. Since \( D \) is negative, the equation has no real roots, indicating the quadratic never crosses the x-axis.

inequalities analysis

When solving quadratic inequalities, the goal is to determine the values of \( x \) that make the inequality true. The steps typically involve:

• Rewrite the inequality in standard form.
• Calculate the discriminant to understand the behavior of the quadratic expression.
• Analyze the sign of the quadratic expression over different intervals of \( x \).

In the given problem, we analyze \(x^2 - x + 1 \leq 0\). We find that the discriminant is negative, so the quadratic doesn’t have real roots and is always positive for all real \(x\). Thus, it isn’t possible for the expression to be less than or equal to zero. Hence, there are no solutions to \(x^2 - x + 1 \leq 0\). Stick to these steps, and you'll handle quadratic inequalities more confidently.