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Chapter 1: Problem 31

Solve. Some of your answers may involve \(i\). \(4 x^{2}=4 x+1\)

Short Answer

Expert verified
\[x = \frac{1 + \sqrt{2}}{2}\] and \[x = \frac{1 - \sqrt{2}}{2}\]

Step by step solution

01

Rewrite the Equation

Rewrite the equation into standard quadratic form: \[4x^2 - 4x - 1 = 0\]
02

Identify Coefficients

Identify the coefficients for the quadratic equation \(ax^2 + bx + c = 0\): Here, \(a = 4\), \(b = -4\), and \(c = -1\).
03

Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\) to find the roots: \[x = \frac{4 \, \pm \, \sqrt{(-4)^2 - 4 \times 4 \times (-1)}}{2 \times 4}\]
04

Simplify Under the Square Root

Simplify the expression under the square root: \[x = \frac{4 \, \pm \, \sqrt{16 + 16}}{8}\]\[x = \frac{4 \, \pm \, \sqrt{32}}{8}\]
05

Simplify the Square Root

Notice that \(\sqrt{32} = \sqrt{16 \times 2} = 4 \sqrt{2}\). Therefore, simplify the equation further: \[x = \frac{4 \, \pm \, 4 \sqrt{2}}{8}\]\[x = \frac{1 \, \pm \, \sqrt{2}}{2}\]
06

Present the Solutions

The solutions to the quadratic equation are: \[x = \frac{1 + \sqrt{2}}{2} \quad \text{and} \quad x = \frac{1 - \sqrt{2}}{2}\]

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Key Concepts

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

quadratic formula
Quadratic equations can be quite intimidating, but the quadratic formula is here to help! This formula is a powerful tool that solves any quadratic equation of the form \(ax^2 + bx + c = 0\). The quadratic formula is: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\] The symbols \(a\), \(b\), and \(c\) represent the coefficients of the equation.
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Most popular questions from this chapter

A key \(n\) notes (including half-notes). above the A above middle C generates a sound wave proportional to $$ y=\sin \left(880 \cdot 2^{n / 12} \pi t\right) $$ where \(t\) is in seconds. Use this equation to solve the following exercises. Find the frequency of the A above middle \(C\) \((n=0)\)

Show that \(\sin 2 t=2 \sin t \cos t .\) (Hint: Let \(s=t\) and use a Sum Identity.)

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Use a calculator to evaluate the following trigonometric functions. $$ \cos 125^{\circ} $$

At the base of the basilar membrane, the fibers are short, stiff, and narrow. At the base, the basilar fibers are most affected by what kind of sound frequencies?
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