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Magazine Chapter 11: Problem 21
Solve. \((5 n+1)^{2}+2(5 n+1)-3=0\)
Short Answer
Expert verified
The solutions for \(n\) are \(n = 0\) and \(n = -\frac{4}{5}\).
Step by step solution
01
Substitution
Let's simplify the problem by substituting \(x = 5n + 1\). This transforms the equation into \(x^2 + 2x - 3 = 0\).
02
Identify the Quadratic Equation
We now have a standard quadratic equation \(x^2 + 2x - 3 = 0\). Our task is to solve for \(x\).
03
Use the Quadratic Formula
The quadratic formula is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For our equation, \(a=1\), \(b=2\), and \(c=-3\).
04
Compute the Discriminant
Calculate the discriminant, \(b^2 - 4ac\): \[ 2^2 - 4 \times 1 \times (-3) = 4 + 12 = 16 \]
05
Solve for x
Plug the values into the quadratic formula: \[ x = \frac{-2 \pm \sqrt{16}}{2 \times 1} \]This results in \(x = \frac{-2 \pm 4}{2}\), leading to two solutions, \(x = 1\) and \(x = -3\).
06
Substitute Back and Solve for n
Recall that \(x = 5n + 1\). For each \(x\), solve for \(n\):- If \(x = 1\), then \(5n + 1 = 1\), so \(5n = 0\) and \(n = 0\).- If \(x = -3\), then \(5n + 1 = -3\), so \(5n = -4\) and \(n = -\frac{4}{5}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Quadratic Formula
The quadratic formula is a powerful tool for finding the roots of any quadratic equation, which typically takes the form \(ax^2 + bx + c = 0\). This formula helps you to find the solutions for \(x\) by utilizing the expression:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula is derived from completing the square method, but it provides a direct solution with easy substitution. Here are steps to effectively use it:
- First, ensure that the equation is in the standard quadratic form.
- Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
- Calculate the discriminant, \(b^2 - 4ac\), which will guide you about the nature of the roots.
- Finally, substitute \(a\), \(b\), and the discriminant into the formula to find the values of \(x\).
Discriminant
The discriminant in a quadratic equation is an important component found within the quadratic formula. It is given by the expression \(b^2 - 4ac\). The discriminant tells us about the nature and number of roots the quadratic equation will have:
- If \(b^2 - 4ac > 0\), the equation has two distinct real roots, which means you'll find two different solutions for \(x\).
- If \(b^2 - 4ac = 0\), there is exactly one real root, which is also known as a repeated or double root.
- If \(b^2 - 4ac < 0\), the equation has no real roots, resulting in two complex roots with imaginary numbers.
Substitution Method
The substitution method is a useful algebraic technique to simplify complex expressions before solving them. This approach involves substituting a part of the equation with a single variable, making it easier to handle. For the given exercise:
- We start with the original equation \((5n+1)^2 + 2(5n+1) - 3 = 0\).
- We perform substitution by setting \(x = 5n + 1\). This transforms the problem to a simpler quadratic form: \(x^2 + 2x - 3 = 0\).
- Solving this quadratic equation using the quadratic formula gives us solutions for \(x\).
- After finding the values of \(x\), substitute back to solve for \(n\).
