Question

Find the real solutions, if any, of each equation. Use the quadratic formula. $$ 2+\frac{8}{x}+\frac{3}{x^{2}}=0 $$

Short answer

The real solutions are \(x = -2 + \frac{\sqrt{10}}{2}\) and \(x = -2 - \frac{\sqrt{10}}{2}\).

Step-by-step solution

- Rewrite the equation

Rewrite the given equation to look like a standard quadratic equation. Multiply through by \(x^2\) to clear the fractions: \[ x^2(2) + x^2\frac{8}{x} + x^2\frac{3}{x^2} = 0 \ 2x^2 + 8x + 3 = 0 \]

- Identify coefficients

Identify the coefficients in the standard quadratic form \( ax^2 + bx + c = 0 \): \( a = 2 \), \( b = 8 \), \( c = 3 \)

- Apply the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Substitute the coefficients into the quadratic formula: \[ x = \frac{-8 \pm \sqrt{8^2 - 4(2)(3)}}{2(2)} \] \= \[ x = \frac{-8 \pm \sqrt{64 - 24}}{4} \]

- Simplify the expression

Simplify the expression under the square root and solve for x: \[ x = \frac{-8 \pm \sqrt{40}}{4} \] Since \( \sqrt{40} = 2 \sqrt{10}\), we have: \[ x = \frac{-8 \pm 2 \sqrt{10}}{4} \] Simplify further to get: \[ x = \frac{-8 + 2 \sqrt{10}}{4} \ or \ x = \frac{-8 - 2 \sqrt{10}}{4} \ \] \[ x = -2 + \frac{\sqrt{10}}{2} \ or \ x = -2 - \frac{\sqrt{10}}{2} \]

- Write the final solution

The real solutions to the quadratic equation are: \[ x = -2 \pm \frac{\sqrt{10}}{2} \]

Key concepts

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree. In mathematical terms, it takes the form:
\( ax^2 + bx + c = 0 \).
Here, 'a', 'b', and 'c' are constants. In our example, we start with a non-standard form: \( 2+\frac{8}{x}+\frac{3}{x^{2}}=0 \).
We need to transform it into the standard quadratic form. By multiplying through by \( x^2 \), we get \( 2x^2 + 8x + 3 = 0 \). Now, it looks like a typical quadratic equation.

Solving Equations

To solve a quadratic equation, we use various methods like factoring, completing the square, or applying the quadratic formula. Here, we rely on the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].
First, identify coefficients. For \( 2x^2 + 8x + 3 = 0 \): \( a = 2 \), \( b = 8 \), and \( c = 3 \).
Plug these values into the formula to find 'x'.
\[ x = \frac{-8 \pm \sqrt{64 - 24}}{4} \].
Simplify inside the square root and the fraction result.

Real Solutions

Real solutions refer to values of 'x' that satisfy the quadratic equation within the real number system. We calculate the discriminant \( b^2 - 4ac \).
If it's positive, there are two distinct real solutions. If zero, one real solution. If negative, no real solutions (complex roots).
Here, \( b^2 - 4ac = 64 - 24 = 40 \), which is positive.
Therefore, the equation \( 2x^2 + 8x + 3 = 0 \) has two real solutions.

Algebra

Algebra is the branch of math dealing with symbols and rules for manipulating these symbols. It's essential for solving equations like our quadratic problem.
It involves:

• Identifying coefficients: 'a', 'b', and 'c'.
• Applying formulas and simplifying expressions.
• Understanding how transformations (like multiplying by \( x^2 \)) help rearrange an equation.

Understanding these basics in algebra helps solve more complex problems efficiently.
Therefore, mastering algebra opens doors to advanced math topics and real-world applications.