Solve each inequality algebraically and write any solution in interval notation. x^2-4 x+2 ≤0

Name: Problem 377, Chapter 9: Intermediate Algebra
Uploaded: 2022-09-29T20:24:31-08:00
Channel: Amelia Hardy
Description: Solve each inequality algebraically and write any solution in interval notation. x^2-4 x+2 ≤0

Solve each inequality algebraically and write any solution...

Key Concepts

Quadratic Inequalities

A quadratic inequality involves a second-degree polynomial being set less than, greater than, or equal to zero. Solving such inequalities requires finding the roots of the quadratic equation and determining the intervals on the number line over which the quadratic expression takes positive or negative values. This analysis helps in establishing the set of values that satisfy the original inequality.

Quadratic Formula

The quadratic formula is a universal method for finding the roots of any quadratic equation of the form ax² + bx + c = 0. It provides a step-by-step way of calculating the solutions by using the coefficients of the quadratic. This formula is particularly useful when the quadratic does not factor neatly, and the nature of the roots also helps in understanding the behavior of the quadratic function in an inequality context.

Discriminant Analysis

The discriminant, given by b² - 4ac in the quadratic formula, reveals important information about the nature of the roots of a quadratic equation. A positive discriminant indicates two distinct real roots, a zero discriminant signals one repeated real root, and a negative discriminant implies no real roots. Knowing the discriminant helps in determining the intervals to test in the inequality.

Interval Notation

Interval notation is a system for writing subsets of the real number line that are defined by an inequality. It provides a concise way to denote a continuous range of values, using brackets to include endpoints when the inequality is non-strict and parentheses when it is strict. This notation is widely used to express the final answer of a solved inequality efficiently.

Transcript

00:01 Here i have the quadratic inequality x squared minus 4x plus 2 is less than or equal to 0.

00:08 So what i'm going to do here is i'm going to start by solving the associated quadratic equation, which is going to get us what are called our critical points.

00:17 So in order to solve this quadratic equation, i'm going to recognize that this is not easily factorable.

00:23 So instead of factoring, i'm going to use a process called completing the square, which is going to involve subtracting the constant over to the other side.

00:30 Side, and then adding a new constant k to both sides.

00:36 Now the constant k is always going to equal b over two squared, where b is the coefficient of our linear term.

00:44 So i have negative 4 over 2 squared, which gives us 4.

00:50 So i have x squared minus 4x plus 4 equals negative 2 plus 4.

00:58 So that is going to give us when i factor the left -hand side, x minus 2 squared equals 2.

01:08 From here, i can square root both sides, which will cancel out my square.

01:15 So i have x -minus 2 equals plus or minus 2, root 2.

01:21 From here, add 2 to both sides, and i get x equals 2 plus or minus root 2.

01:30 Now i'm going to draw these points on a number line.

01:34 So i'm going to estimate their value using a calculator.

01:38 So i have the square root of two, subtracted from two, which gives us 0 .59.

01:47 And then i have the square root of two added to 2, which gives us approximately 3 .41...

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