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Magazine Chapter 6: Problem 12
Solve each quadratic equation by using (a) the factoring method and (b) the method of completing the square. $$ 2 n^{2}+7 n-4=0 $$
Short Answer
Expert verified
The solutions are \(n = \frac{1}{2}\) and \(n = -4\).
Step by step solution
01
Identify the Quadratic Equation
Given the quadratic equation is \(2n^2 + 7n - 4 = 0\). We will first solve it using the factoring method, then by the method of completing the square.
02
Factoring the Quadratic Equation
To factor \(2n^2 + 7n - 4 = 0\), we need to find two numbers whose product is \(2 \times -4 = -8\) and whose sum is \(7\). These numbers are \(8\) and \(-1\). Rewrite the middle term using these numbers: \(2n^2 + 8n - n - 4 = 0\). Factor by grouping: \((2n^2 + 8n) + (-n - 4) = 0\). Factor out the greatest common factors: \(2n(n + 4) - 1(n + 4) = 0\). Now factor out \((n + 4)\): \((2n - 1)(n + 4) = 0\).
03
Solve the Factored Equation
Set each factor equal to zero:1. \(2n - 1 = 0\). Solve for \(n\): \[n = \frac{1}{2}\]2. \(n + 4 = 0\). Solve for \(n\): \[n = -4\]The solutions are \(n = \frac{1}{2}\) and \(n = -4\).
04
Completing the Square: Preparation
To complete the square, rearrange the equation: \(2n^2 + 7n = 4\). Divide all terms by \(2\) to make the coefficient of \(n^2\) equal to 1:\[n^2 + \frac{7}{2}n = 2\].
05
Find the Square Completion Term
Take half of the coefficient of \(n\), which is \(\frac{7}{2}\), and square it: \[\left(\frac{7}{4}\right)^2 = \frac{49}{16}\]. Add and subtract this value to the equation: \[n^2 + \frac{7}{2}n + \frac{49}{16} = 2 + \frac{49}{16}\].
06
Form Perfect Square Trinomial
Rewrite the left side as a completed square:\[(n + \frac{7}{4})^2 = \frac{81}{16}\]. Now take the square root of both sides:\[n + \frac{7}{4} = \pm \frac{9}{4}\].
07
Solve the Completed Square Equation
Solve for \(n\):1. \(n + \frac{7}{4} = \frac{9}{4}\). Solve for \(n\): \[n = \frac{9}{4} - \frac{7}{4} = \frac{2}{4} = \frac{1}{2}\]2. \(n + \frac{7}{4} = -\frac{9}{4}\). Solve for \(n\): \[n = -\frac{9}{4} - \frac{7}{4} = -\frac{16}{4} = -4\]The solutions are \(n = \frac{1}{2}\) and \(n = -4\).
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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. It involves rewriting the quadratic expression in the form of a product of two binomials. For the equation provided, \(2n^2 + 7n - 4 = 0\), we need to find two numbers that multiply to \(2 \times (-4) = -8\) and add up to \(7\). These numbers are \(8\) and \(-1\).
Here's how you proceed:
Here's how you proceed:
- Rewrite the middle term \(+7n\) using \(8\) and \(-1\): \(2n^2 + 8n - n - 4\).
- Group the terms: \((2n^2 + 8n) + (-n - 4)\).
- Factor out the common factors: \(2n(n + 4) - 1(n + 4)\).
- Now factor by grouping: \((2n - 1)(n + 4) = 0\).
- \(2n - 1 = 0\) gives \(n = \frac{1}{2}\).
- \(n + 4 = 0\) gives \(n = -4\).
Completing the Square
Completing the square is another method used to solve quadratic equations by converting the equation into a perfect square trinomial. It involves rearranging and manipulating the equation until it can be written as the square of a binomial. For \(2n^2 + 7n - 4 = 0\), start by preparing the equation for completing the square:
Begin by isolating the \(n^2\) and \(n\)-terms:
Begin by isolating the \(n^2\) and \(n\)-terms:
- First, move \(-4\) to the other side: \(2n^2 + 7n = 4\).
- Divide each term by \(2\) to normalize the \(n^2\) coefficient: \(n^2 + \frac{7}{2}n = 2\).
- Take half of \(\frac{7}{2}\) which is \(\frac{7}{4}\), then square it to get \(\frac{49}{16}\).
- Add and subtract this value: \(n^2 + \frac{7}{2}n + \frac{49}{16} = 2 + \frac{49}{16}\).
- The equation becomes \((n + \frac{7}{4})^2 = \frac{81}{16}\).
- \(n + \frac{7}{4} = \pm \frac{9}{4}\).
- This gives two solutions: \(n = \frac{1}{2}\) and \(n = -4\).
Solution Techniques for Quadratics
Quadratic equations, typically in the form \(ax^2 + bx + c = 0\), have several methods available for finding solutions. Understanding these methods can provide flexibility depending on the complexity of the quadratic equation. Common solution techniques include:
- Factoring: Useful when the quadratic can be easily rewritten as a product of binomials. It is direct and reveals roots that satisfy the equation.
- Completing the Square: Converts the quadratic into a perfect square trinomial. This method is beneficial when factoring is difficult or when transformations are needed to apply the quadratic formula.
- Quadratic Formula: An all-encompassing solution \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It is versatile and works when factoring is not straightforward or other methods are cumbersome.
- Graphing: Visualizes the solutions. The roots are where the graph of the quadratic equation intersects the x-axis.
