/*! 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 62 \(n^{2}=10 n+8\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 62

\(n^{2}=10 n+8\)

Short Answer

Expert verified
\(n = 5 + \sqrt{33}\) or \(n = 5 - \sqrt{33}\)

Step by step solution

01

Set the Equation to Zero

Rewrite the given equation by bringing all terms to one side to set it to zero: \(n^{2} - 10n - 8 = 0\)
02

Identify Coefficients

Identify the coefficients from the quadratic equation in standard form \(an^{2} + bn + c = 0\). Here, \(a = 1\), \(b = -10\), and \(c = -8\).
03

Use the Quadratic Formula

The quadratic formula is \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the identified coefficients into this formula.
04

Calculate the Discriminant

Calculate the discriminant, \(b^2 - 4ac\): \((-10)^2 - 4(1)(-8) = 100 + 32 = 132\).
05

Compute the Solutions

Substitute the discriminant back into the quadratic formula to find the solutions for \(n\): \(n = \frac{10 \pm \sqrt{132}}{2} = \frac{10 \pm 2\sqrt{33}}{2} = 5 \pm \sqrt{33}\).

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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 solved using the quadratic formula, which is a powerful tool. It helps us find the values of the variable that make the equation true. The quadratic formula is: \[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here, the symbols have specific meanings:By plugging in these values from any quadratic equation, we can find the solutions, known as the roots of the equation. The formula derives from the process of completing the square and works for all quadratic equations.
Discriminant Calculation
The discriminant is a key part of the quadratic formula that helps determine the nature of the roots of the quadratic equation. It is calculated using: \[ b^2 - 4ac \]
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
  • If the discriminant is zero, the quadratic equation has exactly one real root, also known as a repeated or double root.
  • If the discriminant is negative, the quadratic equation has two complex roots.
Through the discriminant, we can understand if the solutions are real or imaginary before actually calculating the roots.
Standard Form of Quadratic Equation
A quadratic equation is usually expressed in the standard form: \[ax^2 + bx + c = 0\]To use the quadratic formula effectively, the equation must be in this form. Here鈥檚 how to convert any given quadratic equation to standard form:
  • Move all terms to one side of the equation
  • Arrange them in descending order of their degrees
For example, for the equation \(n^2 = 10n + 8\), we subtract \(10n\) and 8 from both sides to get:\[n^2 - 10n - 8 = 0\]Now it鈥檚 in the standard form, ready for applying the quadratic formula to find the solutions.

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Most popular questions from this chapter

\(y=x^{2}-10 x+29\)

. Describe the difference between a rational number and an irrational number.

The diameter at basal height of a tree is the diameter about \(3 \mathrm{ft}\) above the ground. The approximate shape of the trunk of a tree above basal height is a cone. The formula for the volume of a cone is \(V=\frac{\pi r^{2} h}{3}\), where \(r\) is the radius and \(h\) is the height. Measured from its basal height, a tree is \(50 \mathrm{ft}\) tall. The diameter of its trunk is \(2.5 \mathrm{ft}\). Find the approximate volume of lumber in cubic feet in this trunk. ( \(\pi \approx 3.14\).) Round to the nearest whole number. (Source: G. John Smith; www.math.bcit.ca, 1997)

\(n^{2}+5 n+6=0\)

\(2 n^{2}+6=-20\)
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