Solve each quadratic inequality. Give the solution set in interval notation. See Exam-ples 5 and 6. x^2-2x≤1

Verified step by step guidance

1. Rewrite the inequality in standard form: $x^2-2x-1\le 0$.
2. Find the roots of the corresponding quadratic equation $x^2-2x-1=0$ using the quadratic formula: $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$, where $a=1$, $b=-2$, and $c=-1$.
3. Calculate the discriminant $b^2-4ac$ to determine the nature of the roots.
4. Use the roots to divide the number line into intervals and test a point from each interval in the inequality $x^2-2x-1\le 0$ to determine where the inequality holds true.
5. Express the solution set in interval notation based on the intervals where the inequality is satisfied.

Key Concepts

Quadratic Inequalities

Quadratic inequalities are expressions that involve a quadratic polynomial set in relation to a value, typically using symbols like ≤, ≥, <, or >. To solve these inequalities, one must first rearrange the inequality into standard form, then find the roots of the corresponding quadratic equation, and finally determine the intervals where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Test Points

Test points are specific values chosen from the intervals created by the roots of a quadratic inequality. By substituting these test points back into the original inequality, one can determine whether the inequality is satisfied in that interval. This method helps to identify the solution set effectively by confirming which intervals meet the inequality's conditions.