Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 10: Problem 38
\(n^{2}+9 n-36=0\)
Short Answer
Expert verified
The roots are \(n = 3\) and \(n = -12\).
Step by step solution
01
Identify the Type of Equation
Notice that the equation \(n^2 + 9n - 36 = 0\) is a quadratic equation because it is in the standard form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 9\), and \(c = -36\).
02
Calculate the Discriminant
The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula \(b^2 - 4ac\). Plug in the values \(a = 1\), \(b = 9\), and \(c = -36\) to calculate:\[\Delta = 9^2 - 4(1)(-36) = 81 + 144 = 225.\]
03
Determine the Nature of the Roots
Since the discriminant \(\Delta = 225\) is a positive perfect square, the quadratic equation has two distinct real roots.
04
Apply the Quadratic Formula
Use the quadratic formula \(n = \frac{{-b \pm \sqrt{\Delta}}}{{2a}}\) to find the roots. Substitute \(b = 9\), \(\Delta = 225\), and \(a = 1\):\[n = \frac{{-9 \pm \sqrt{225}}}{2(1)} = \frac{{-9 \pm 15}}{2}.\]
05
Calculate the Roots
Now compute the values of \(n\):1. \(n_1 = \frac{{-9 + 15}}{2} = \frac{6}{2} = 3\)2. \(n_2 = \frac{{-9 - 15}}{2} = \frac{-24}{2} = -12\)Therefore, the roots of the equation are \(n = 3\) and \(n = -12\).
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a reliable method for finding the roots of any quadratic equation. Quadratic equations have the general form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known values, with \(a eq 0\). For our original equation \(n^2 + 9n - 36 = 0\), the values are \(a = 1\), \(b = 9\), and \(c = -36\).The quadratic formula is written as:\[x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}\]To find the roots using this formula:
- Substitute \(b = 9\), \(a = 1\), and \(c = -36\) into the formula.
- Calculate the discriminant, which we'll discuss next.
- Solve for the roots using the derived values.
Discriminant
The discriminant is a key part of the quadratic formula and is crucial in determining the nature of the roots of a quadratic equation. It is represented by \(\Delta\) and defined as:\[\Delta = b^2 - 4ac\]For the equation \(n^2 + 9n - 36 = 0\), with \(a = 1\), \(b = 9\), and \(c = -36\), the discriminant is calculated as:\[\Delta = 9^2 - 4 \times 1 \times (-36) = 81 + 144 = 225\]The discriminant tells us about the nature of the roots:
- If \(\Delta > 0\), there are two distinct real roots.
- If \(\Delta = 0\), the roots are real and equal (one root repeated).
- If \(\Delta < 0\), the roots are non-real (complex roots).
Real Roots
Real roots are solutions to the quadratic equation that can be found on the real number line. When solving the quadratic equation \(n^2 + 9n - 36 = 0\) with the quadratic formula, we encountered two distinct real roots. This outcome was anticipated because the discriminant was positive, signaling multiple real roots.To find these roots, recall the calculation:
- With the quadratic formula, \(n = \frac{{-b \pm \sqrt{\Delta}}}{2a}\).
- Using \(b = 9\), \(a = 1\), and \(\Delta = 225\), we solvde for \(n\):
- \(n_1 = \frac{{-9 + 15}}{2} = \frac{6}{2} = 3\)
- \(n_2 = \frac{{-9 - 15}}{2} = \frac{-24}{2} = -12\)
