Problem 39 Use the quadratic formula to sol... [FREE SOLUTION]

Chapter 8: Problem 39

Use the quadratic formula to solve each equation. These equations have real solutions and complex but not real solutions. See Examples 1 through 4. $$ (n-2)^{2}=2 n $$

Short Answer

Expert verified

The solutions are $n = 3 + \sqrt{5}$ and $n = 3 - \sqrt{5}$.

Step by step solution

Expand the Equation

First, expand the left-hand side of the equation. The expression $(n-2)^2$ expands to $n^2 - 4n + 4$. So the equation becomes: \[ n^2 - 4n + 4 = 2n \]

Rearrange the Equation

Next, rearrange the equation to bring all terms to one side, resulting in a standard quadratic form. Subtract $2n$ from both sides: \[ n^2 - 4n + 4 - 2n = 0 \] Simplify the equation to: \[ n^2 - 6n + 4 = 0 \]

Identify Coefficients

Identify the coefficients of the quadratic equation $n^2 - 6n + 4 = 0$. Here, $a = 1$, $b = -6$, and $c = 4$.

Apply the Quadratic Formula

Use the quadratic formula $n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ to find the roots of the equation. Substitute $a = 1$, $b = -6$, and $c = 4$ into the formula: \[ n = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 4}}{2 \cdot 1} \]

Simplify Under the Square Root

Calculate the expression under the square root, known as the discriminant. \[ b^2 - 4ac = (-6)^2 - 4 \cdot 1 \cdot 4 = 36 - 16 = 20 \] So, the discriminant is 20.

Simplify the Solutions

Substitute the discriminant back into the quadratic formula and simplify: \[ n = \frac{6 \pm \sqrt{20}}{2} \] Simplifying further, since $\sqrt{20} = 2\sqrt{5}$: \[ n = \frac{6 \pm 2\sqrt{5}}{2} \] The final solutions are: \[ n = 3 \pm \sqrt{5} \]

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Key Concepts

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

Discriminant

The discriminant is a crucial part of solving quadratic equations. It's the part under the square root in the quadratic formula: $ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $. Here, $ b^2 - 4ac $ is the discriminant.
The value of the discriminant determines the nature of the roots of the quadratic equation:

If it's positive, the equation has two distinct real roots.
If it's zero, the equation has exactly one real root, known as a double root.
If it's negative, the equation gives complex solutions with no real roots.

In the exercise, the discriminant value is 20, which is positive. This means the quadratic equation $ n^2 - 6n + 4 = 0 $ has two distinct real solutions. Knowing what the discriminant tells you about the roots can help you anticipate the kind of solutions to expect without solving the entire equation.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree. They typically look like this: $ ax^2 + bx + c = 0 $, where $ a $, $ b $, and $ c $ are constants, and $ a eq 0 $. In our exercise, the standard form of the given equation is $ n^2 - 6n + 4 = 0 $.
To solve them, we use the quadratic formula mentioned earlier. But before that, we often rearrange the original equation into this standard form. A quadratic equation can also be solved by:

Factoring, when it is possible and easy.
Completing the square, which is another systematic method.
Using graphs to estimate solutions.

However, the quadratic formula is always applicable and reliable, especially when factoring isn鈥檛 straightforward. This approach gives both real and complex solutions.

Complex Solutions

If the discriminant is negative, the equation will result in complex solutions. Complex numbers are formed by a real part and an imaginary part: $ a + bi $; where $ i $ is the imaginary unit with the property $ i^2 = -1 $.
Real solutions exist when the discriminant is zero or positive as in our exercise. In contrast, a negative discriminant results in imaginary solutions, changing the equation outcome significantly. Here鈥檚 how it works:

When roots are complex, you'll see the use of $ i $, showing real parts and imaginary parts, indicating there are no real number solutions.
These complex solutions are conjugates of each other for any quadratic equation and are represented as $ a + bi $ and $ a - bi $.

Understanding complex numbers helps in fields like physics and engineering where non-real solutions are often necessary to model systems accurately.

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91影视

Use the quadratic formula to solve each equation. These equations have real solutions and complex but not real solutions. See Examples 1 through 4. $$ (n-2)^{2}=2 n $$

Short Answer

Step by step solution

Expand the Equation

Rearrange the Equation

Identify Coefficients

Apply the Quadratic Formula

Simplify Under the Square Root

Simplify the Solutions

Key Concepts

Discriminant

Quadratic Equations

Complex Solutions

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