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Magazine Chapter 9: Problem 80
Solve by completing the square. $$n^{2}-2 n=-3$$
Short Answer
Expert verified
n = 1 \pm i \sqrt{2}
Step by step solution
01
- Move the constant to the right side
Move the constant term (-3) to the right side of the equation to have the quadratic and linear terms on the left side. The equation becomes: \[ n^{2} - 2n + 3 = 0 \]
02
- Prepare to complete the square
In order to complete the square, we need to form a perfect square trinomial on the left side of the equation. First, take half of the coefficient of the linear term (-2), square it, and add it to both sides. Half of -2 is -1, and squaring it gives 1. So, add 1 to both sides: \[ n^{2} - 2n + 1 = -3 + 1 \] which simplifies to: \[ n^{2} - 2n + 1 = -2 \]
03
- Write the left side as a square
The left side of the equation \( n^{2} - 2n + 1 \) is now a perfect square trinomial. It can be factored as: \[ (n - 1)^{2} = -2 \]
04
- Solve the equation
To isolate \( n \), take the square root of both sides. \[ \text{Since we have} (n - 1)^{2} = -2, \text{taking the square root of both sides gives:} \] \[ n - 1 = \pm \sqrt{-2} \] \( \sqrt{-2} \) can be written as \( \pm i \sqrt{2} \) where \( i \) is the imaginary unit. So, we have: \[ n - 1 = \pm i \sqrt{2} \]
05
- Solve for n
Finally, isolate \( n \) by adding 1 to both sides: \[ n = 1 \pm i \sqrt{2} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Completing the Square
Completing the square is a method used to solve quadratic equations by transforming them into a perfect square trinomial. This approach is handy because it allows us to solve quadratics that are not easily factorable. Here's a quick overview of how it works:
- Start with a quadratic equation in the form of \(ax^2 + bx + c = 0\).
- Move the constant term to the right side of the equation.
- To form a perfect square trinomial on the left side, take half of the coefficient of the linear term (\(bx\)), square it, and add it to both sides of the equation.
- Factor the left side as a perfect square trinomial.
- Finally, take the square root of both sides and solve for the variable.
Imaginary Numbers
Imaginary numbers are a type of number that extends the real number system. They are used when real solutions do not exist for equations, like when you take the square root of a negative number. The imaginary unit is denoted as \(i\), where \(i^2 = -1\).Here's how we incorporate imaginary numbers:
- When you encounter a square root of a negative number, you express it using \(i\). For example, \(\sqrt{-4} = 2i\).
- This is because \(\sqrt{-4}\) can be broken down into \(\sqrt{4} \times \sqrt{-1}\). Since \(\sqrt{-1} = i\), this simplifies to \(2i\).
Solving Quadratic Equations
Solving quadratic equations involves finding the values of the variable that satisfy the equation \(ax^2 + bx + c = 0\). There are multiple methods to solve quadratics, including factoring, using the quadratic formula, and completing the square (as demonstrated here). When using the quadratic formula, we use the equation \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Here's a summary of the process to solve quadratics:
- For factoring, we rewrite the quadratic as a product of binomials (if possible).
- Using the quadratic formula involves plugging the coefficients into the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
- Completing the square, as we've done in this exercise, involves forming a perfect square trinomial and solving for the variable.
