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Magazine Chapter 6: Problem 9
Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$ 2 n^{2}-n-15=0 $$
Short Answer
Expert verified
The solutions are \(n = 3\) and \(n = -\frac{5}{2}\).
Step by step solution
01
Write the equation in standard form
The given quadratic equation is already in standard form: \(2n^2 - n - 15 = 0\). This form is \(ax^2 + bx + c = 0\) where \(a = 2\), \(b = -1\), and \(c = -15\).
02
Factoring method - Find two numbers
For the method of factoring, we need two numbers that multiply to give \(a \cdot c = 2 \times (-15) = -30\) and add to give \(b = -1\). These numbers are 5 and -6.
03
Factoring method - Rewrite middle term
Rewrite the middle term of the equation using the numbers found: \(2n^2 + 5n - 6n - 15 = 0\).
04
Factoring by grouping
Group the terms to factor them separately: \((2n^2 + 5n) + (-6n - 15) = 0\). Factoring each group gives \(n(2n + 5) - 3(2n + 5) = 0\).
05
Factor out common binomial
The expression can be factored as \((n - 3)(2n + 5) = 0\). This implies that either \(n - 3 = 0\) or \(2n + 5 = 0\).
06
Solve using the factoring method
Solve each equation: \(n - 3 = 0\) gives \(n = 3\) and \(2n + 5 = 0\) gives \(n = -\frac{5}{2}\). These are the solutions using the factoring method.
07
Completing the Square - Make leading coefficient 1
If the leading coefficient is not 1, divide the entire equation by 2: \(n^2 - \frac{1}{2}n - \frac{15}{2} = 0\).
08
Move constant to the right side
Move the constant term to the right side: \(n^2 - \frac{1}{2}n = \frac{15}{2}\).
09
Find the square term
To complete the square, take half of the coefficient of \(n\), square it, and add to both sides: \((-\frac{1}{4})^2 = \frac{1}{16}\). Add this to both sides to get \(n^2 - \frac{1}{2}n + \frac{1}{16} = \frac{15}{2} + \frac{1}{16}\).
10
Simplify and express as a square
The left side becomes a perfect square: \((n - \frac{1}{4})^2\). Simplify the right side: \(\frac{15}{2} + \frac{1}{16} = \frac{120}{16} + \frac{1}{16} = \frac{121}{16}\).
11
Solve the equation
Now solve \((n - \frac{1}{4})^2 = \frac{121}{16}\) by taking the square root of both sides: \(n - \frac{1}{4} = \pm \frac{11}{4}\). Solve for \(n\) to get \(n = \frac{1}{4} + \frac{11}{4} = 3\) and \(n = \frac{1}{4} - \frac{11}{4} = -\frac{10}{4} = -\frac{5}{2}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Factoring Method
The factoring method is a popular technique to solve quadratic equations by expressing them as a product of simpler binomial factors. It's particularly useful when the quadratic equation can be factored without complex numbers.
Here's a simple way to understand it:
Here's a simple way to understand it:
- Start by writing the equation in standard form: \( ax^2 + bx + c = 0 \).
- Identify coefficients \(a\), \(b\), and \(c\).
- Find two numbers that multiply to \( a \cdot c \) and add to \( b \).
- Use these numbers to rewrite the middle term, allowing you to group and factor the equation.
- The numbers 5 and -6 work as they multiply to -30 (\(a \times c\)) and add to -1 (\(b\)).
- Rewrite as: \(2n^2 + 5n - 6n - 15 = 0\).
- Group and factor: \(n(2n + 5) - 3(2n + 5) = 0\).
Completing the Square
Completing the square is a method that can always solve a quadratic equation by rewriting it in a form where the left side of the equation is a perfect square trinomial.
Here's how it works:
Here's how it works:
- Ensure the quadratic term has a coefficient of 1. If not, divide the entire equation by \(a\).
- Move the constant term to the other side of the equation.
- Find and add \(\left( \frac{b}{2} \right)^2\) to both sides to form a perfect square trinomial.
- Divide by 2: \(n^2 - \frac{1}{2}n - \frac{15}{2} = 0\).
- Move the constant: \(n^2 - \frac{1}{2}n = \frac{15}{2}\).
- Add \(\left(-\frac{1}{4}\right)^2 = \frac{1}{16}\) to both sides.
- The equation becomes: \(n^2 - \frac{1}{2}n + \frac{1}{16} = \frac{121}{16}\).
- The left side is the square \((n - \frac{1}{4})^2\).
Solution Methods
Quadratic equations can be solved using several methods, each with its own advantages. Understanding these methods allows flexibility when tackling any equation.
Common solution methods include:
Common solution methods include:
- Factoring Method: Fast and effective when equations can be easily reduced to simple factors.
- Completing the Square: A foolproof method that works for all quadratic equations, although sometimes requires dealing with fractions.
- Quadratic Formula: Offers a direct solution using \(n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\); perfect when factoring is not straightforward.
- Graphical Method: Involves plotting the quadratic function and finding the \(x\)-intercepts, giving a visual perspective of the solutions.
