Solve each quadratic inequality. Give the solution set in interval notation. x²-2x≤1

Verified step by step guidance

1. Rewrite the inequality in standard quadratic form by moving all terms to one side: \(x^{2} - 2x - 1 \leq 0\).
2. Identify the quadratic expression: \(x^{2} - 2x - 1\) and set it equal to zero to find the boundary points: \(x^{2} - 2x - 1 = 0\).
3. Use the quadratic formula to solve for \(x\): \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=-2\), and \(c=-1\).
4. Calculate the roots from the quadratic formula to find the critical points that divide the number line into intervals.
5. Test values from each interval in the inequality \(x^{2} - 2x - 1 \leq 0\) to determine where the inequality holds true, then express the solution set in interval notation.

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a quadratic expression set less than, greater than, or equal to a value. Solving it requires finding the range of x-values that satisfy the inequality, often by analyzing the related quadratic equation and its graph.

Solving Quadratic Equations

To solve a quadratic inequality, first solve the corresponding quadratic equation by setting it equal to zero. This helps find critical points (roots) that divide the number line into intervals to test for the inequality.

Interval Notation and Testing Intervals

After finding the roots, the number line is split into intervals. Test points from each interval in the inequality to determine where it holds true. Express the solution set using interval notation, which concisely represents all valid x-values.