/*! 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 \(n^{2}-8 n-48=0\)... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 40

\(n^{2}-8 n-48=0\)

Short Answer

Expert verified
The solutions are \(n = 12\) and \(n = -4\).

Step by step solution

01

Identify the Type of Equation

The given equation is a quadratic equation in the standard form of \(an^{2} + bn + c = 0\), where \(a = 1\), \(b = -8\), and \(c = -48\).
02

Apply the Quadratic Formula

To solve the equation \(n^{2} - 8n - 48 = 0\), we can use the quadratic formula: \(n = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\).
03

Calculate the Discriminant

The discriminant \(b^2 - 4ac\) is calculated as follows: \((-8)^2 - 4(1)(-48) = 64 + 192 = 256\). The discriminant is positive, indicating two real roots.
04

Solve for the Roots

Plug the values into the quadratic formula: \[ n = \frac{-(-8) \pm \sqrt{256}}{2(1)} \] This simplifies to: \[ n = \frac{8 \pm 16}{2} \]Thus, the roots are \(n = \frac{24}{2} = 12\) and \(n = \frac{-8}{2} = -4\).
05

Verify the Solution

Check the values by substituting them back into the original equation. For \(n = 12\), \(12^{2} - 8(12) - 48 = 144 - 96 - 48 = 0\). For \(n = -4\), \((-4)^{2} - 8(-4) - 48 = 16 + 32 - 48 = 0\). Thus, both are solutions.

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Key Concepts

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

Quadratic Formula
When you're dealing with a quadratic equation like the one given \(n^2 - 8n - 48 = 0\), a powerful tool is the quadratic formula. This formula helps you find the roots (solutions) of any quadratic equation and is expressed as:\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here is a breakdown of the elements:
  • \(a\), \(b\), and \(c\) are coefficients from the quadratic equation in the standard form \(ax^2 + bx + c = 0\).
  • The "\(\pm\)" symbol means you'll be finding two potential solutions.
To use the formula, you must first identify \(a\), \(b\), and \(c\) from your equation. In our case, \(a = 1\), \(b = -8\), and \(c = -48\). Then plug these into the formula, and don鈥檛 forget to compute the discriminant \(b^2 - 4ac\) as it's crucial in determining the nature of the roots.
Discriminant
The discriminant in a quadratic equation, represented by \(b^2 - 4ac\), helps determine the nature of the roots without having to solve the entire equation. It can reveal whether the equation has:
  • Two distinct real roots
  • One real double root
  • Two complex roots
For the equation \(n^2 - 8n - 48 = 0\), let's compute the discriminant:\[ b^2 - 4ac = (-8)^2 - 4 \times 1 \times (-48) = 64 + 192 = 256 \]A positive discriminant (like 256 in this example) tells us there are two distinct real roots. If the discriminant had been zero, it would mean a single real root (or two identical roots). A negative discriminant signifies complex roots. Knowing the discriminant gives you a quick insight into the type of solutions you can expect.
Roots of a Quadratic Equation
The roots of a quadratic equation are the values of \(x\) (or in our case, \(n\)) that make the equation true Here, after computing using the quadratic formula, we found:\[ n = \frac{-(-8) \pm \sqrt{256}}{2} \]which simplifies to:\[ n = \frac{8 \pm 16}{2} \]This gives us two roots:
  • \(n = \frac{24}{2} = 12\)
  • \(n = \frac{-8}{2} = -4\)
These values of \(n\) are the points where the parabola described by the equation \(n^2 - 8n - 48 = 0\) intersects the \(n\)-axis. Each root represents a solution to the quadratic equation. To verify, you can substitute these roots back into the original equation, as was done in the solution steps. Both will result in zero, confirming their validity as the correct roots.

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