/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 40 \(n^{2}+10 n+2=0\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 40

\(n^{2}+10 n+2=0\)

Short Answer

Expert verified
\( n = -5 \pm \sqrt{23} \)

Step by step solution

01

Identify the equation

Given the quadratic equation in standard form: \[n^2 + 10n + 2 = 0\]
02

Calculate the discriminant

The discriminant \( \text{螖} \) of a quadratic equation \(ax^2 + bx + c = 0\) is calculated using the formula: \[ \text{螖} = b^2 - 4ac \] Here, \( a = 1 \), \( b = 10 \), and \( c = 2 \). So, \[ \text{螖} = 10^2 - 4 \times 1 \times 2 = 100 - 8 = 92 \]
03

Apply the quadratic formula

The solutions to the quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula: \[ n = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \] Substitute the values of \(a\), \(b\), and \(c\): \[ n = \frac{-10 \, \pm \, \sqrt{92}}{2 \times 1} = \frac{-10 \, \pm \, \sqrt{92}}{2} \]
04

Simplify the expression

Simplify under the square root if possible, then solve for the two roots: \[ \sqrt{92} = \sqrt{4 \times 23} = 2\sqrt{23} \] So, \[ n = \frac{-10 \, \pm \, 2\sqrt{23}}{2} = -5 \, \pm \, \sqrt{23} \]
05

State the solutions

The solutions for the equation \(n^2 + 10n + 2 = 0\) are: \[ n = -5 + \sqrt{23} \] and \[ n = -5 - \sqrt{23} \]

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Key Concepts

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

Discriminant Calculation
To solve a quadratic equation, we first need to determine its discriminant, denoted as \(\text{螖}\). The discriminant helps us understand the nature of the solutions (roots) for the quadratic equation. The general form for a quadratic equation is \(ax^2 + bx + c = 0\). The discriminant is calculated using the formula: \(\text{螖} = b^2 - 4ac\).

In our specific example, the quadratic equation is \(n^2 + 10n + 2 = 0\). Here:
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