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Chapter 9: Problem 377

Solve each inequality algebraically and write any solution in interval notation. $$x^{2}-4 x+2 \leq 0$$

Short Answer

Expert verified
[2 - \(\sqrt{2}\), 2 + \(\sqrt{2}\)]

Step by step solution

01

Write the Inequality in Standard Quadratic Form

The inequality is already in standard form: \[x^{2} - 4x + 2 \leq 0\]
02

Solve the Corresponding Quadratic Equation

Solve the equation \(x^{2} - 4x + 2 = 0\). Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -4\), and \(c = 2\).\[x = \frac{4 \pm \sqrt{16 - 8}}{2} = \frac{4 \pm \sqrt{8}}{2} = \frac{4 \pm 2\sqrt{2}}{2} = 2 \pm \sqrt{2}\]
03

Determine the Critical Points

From the quadratic formula, the critical points are \(2 + \sqrt{2}\) and \(2 - \sqrt{2}\). These points divide the number line into three intervals.
04

Test Intervals Around Critical Points

Test points from each interval: 1. For interval \((-\infty, 2 - \sqrt{2})\), pick \(x = 0\): \[x^{2} - 4x + 2 = 0 - 0 + 2 = 2 \ (> 0)\]2. For interval \((2 - \sqrt{2}, 2 + \sqrt{2})\), pick \(x = 2\): \[x^{2} - 4x + 2 = 4 - 8 + 2 = -2 \ (\leq 0)\]3. For interval \((2 + \sqrt{2}, \infty)\), pick \(x = 3\): \[x^{2} - 4x + 2 = 9 - 12 + 2 = -1 \ (> 0) \]
05

Write the Solution in Interval Notation

The inequality is satisfied in the interval \((2 - \sqrt{2}, 2 + \sqrt{2})\). Include the critical points as they satisfy the inequality. The solution in interval notation is \([2 - \sqrt{2}, 2 + \sqrt{2}]\).

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Key Concepts

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

Quadratic Equations
A quadratic equation is an equation of the form

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Most popular questions from this chapter

(a) graph the quadratic functions on the same rectangular coordinate system and (b) describe what effect adding a constant, \(h\), inside the parentheses has \(f(x)=x^{2}, g(x)=(x+4)^{2}\) and \(h(x)=(x-4)^{2}\)

Graph each function. $$f(x)=\frac{1}{4} x^{2}$$

Graph each function using a vertical shift. $$f(x)=x^{2}-7$$

(a) rewrite each function in \(f(x)=a(x-h)^{2}+k\) form and (b) graph it by using transformations. $$f(x)=3 x^{2}-6 x-1$$

Graph each function using transformations. $$f(x)=(x+5)^{2}-2$$
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