Problem 14 Simplify and reduce each express... [FREE SOLUTION]

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Intermediate Algebra

Jerome E. Kaufmann, Karen L. Schwitters

$Math Studyset 91影视 Explanations$ Math

9 Edition

Chapter 6: Problem 14

Simplify and reduce each expression. $$ n^{2}+3 n-2=0 $$

Short Answer

Expert verified

The solutions for $n$ are $n = \frac{{-3 + \sqrt{17}}}{2}$ and $n = \frac{{-3 - \sqrt{17}}}{2}$.

Step by step solution

Understanding the Equation

The given quadratic equation is $n^2 + 3n - 2 = 0$. Our objective is to find the values of $n$ that satisfy this equation.

Identifying the Quadratic Formula

For a quadratic equation of the form $an^2 + bn + c = 0$, the quadratic formula is $n = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a}$. In this equation, $a = 1$, $b = 3$, and $c = -2$.

Substitute Values into the Quadratic Formula

Plug the values for $a$, $b$, and $c$ into the quadratic formula: $n = \frac{{-3 \pm \sqrt{{3^2 - 4(1)(-2)}}}}{2(1)}$.

Calculating the Discriminant

The discriminant $b^2 - 4ac$ is calculated as $3^2 - 4 \times 1 \times (-2) = 9 + 8 = 17$.

Solving for n

Using the discriminant, we solve for $n$ as follows: $n = \frac{{-3 \pm \sqrt{17}}}{2}$. This gives us two potential solutions for $n$, namely $n = \frac{{-3 + \sqrt{17}}}{2}$ and $n = \frac{{-3 - \sqrt{17}}}{2}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Formula

When dealing with quadratic equations, specifically those of the form $an^2 + bn + c = 0$, the quadratic formula is a critical tool. It provides an efficient method to find the solutions of the equation. The formula is expressed as:

$n = \frac{{-b \pm \sqrt{b^2 - 4ac}}}{2a}$

This formula directly calculates the values of $n$, which are the points where the quadratic equation equals zero, also known as the roots.
The different components of this formula are:

$a$: Coefficient of $n^2$
$b$: Coefficient of $n$
$c$: Constant term

By substituting the coefficients from your specific equation, you can solve for $n$, ensuring you find all potential solutions of the quadratic equation.

Discriminant

The discriminant is a pivotal part of the quadratic formula, determining the nature and number of the roots. It is denoted as $b^2 - 4ac$.
Here鈥檚 what the discriminant tells us:

If $b^2 - 4ac > 0$, the quadratic equation has two distinct real solutions.
If $b^2 - 4ac = 0$, there is exactly one real solution, a repeated root.
If $b^2 - 4ac < 0$, the equation has no real solutions but two complex ones.

In our example equation $n^2 + 3n - 2 = 0$, calculating the discriminant $(3^2 - 4 \times 1 \times (-2))$ gives us 17, indicating two distinct real roots due to its positive value.

Equation Solving

Solving quadratic equations revolves around finding values for the variable that satisfy the equation.
Once you've identified the coefficients $a$, $b$, and $c$, and calculated the discriminant, you can proceed with the quadratic formula to find the solutions.
In our example, with a discriminant of 17, solutions are determined by the formula:

$n = \frac{{-3 \pm \sqrt{17}}}{2}$

This results in two solutions:

$n = \frac{{-3 + \sqrt{17}}}{2}$
$n = \frac{{-3 - \sqrt{17}}}{2}$

These solutions represent the values of $n$ where the quadratic equation equals zero, offering a complete answer to the original problem.

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91影视

Simplify and reduce each expression. $$ n^{2}+3 n-2=0 $$

Short Answer

Step by step solution

Understanding the Equation

Identifying the Quadratic Formula

Substitute Values into the Quadratic Formula

Calculating the Discriminant

Solving for n

Key Concepts

Quadratic Formula

Discriminant

Equation Solving

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