Problem 43 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. \(31-44\) : Solve by using the quadratic formula. $$ \frac{5 x}{x^{2}+9}=-1 $$

Step-by-Step Solution

Answer

The equation has no real solutions; the solutions are complex numbers.

Step 1: Clear the Fraction

To eliminate the fraction from the equation, multiply both sides by \(x^2 + 9\):\[5x = -1(x^2 + 9)\]This simplifies to:\[5x = -x^2 - 9\]

Step 2: Rearrange into Standard Form

Next, rearrange the equation to form a standard quadratic equation \(ax^2 + bx + c = 0\):\[x^2 + 5x + 9 = 0\]

Step 3: Identify Coefficients

Identify the coefficients of the quadratic equation:- \(a = 1\)- \(b = 5\)- \(c = 9\)

Step 4: Apply the Quadratic Formula

The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Substitute the coefficients into the formula:\[x = \frac{-5 \pm \sqrt{5^2 - 4 \cdot 1 \cdot 9}}{2 \cdot 1}\]{"

Step 5: Calculate the Discriminant

Calculate the discriminant (\(b^2 - 4ac\)):\[5^2 - 4 \cdot 1 \cdot 9 = 25 - 36 = -11\]

Step 6: Interpret the Discriminant

Since the discriminant is negative, \(-11\), this indicates there are no real solutions to the equation. The solutions are complex numbers.

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool in algebra used to find the solutions of quadratic equations. The general form of a quadratic equation is \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients with \(a eq 0\). The quadratic formula solves for \(x\) by:

• Substituting the coefficients \(a\), \(b\), and \(c\) into the formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).
• The symbol \(\pm\) denotes that there are potentially two solutions: one using the plus sign and the other using the minus.

By applying this formula, you can systematically find the roots of any quadratic equation. This formula works with all quadratic equations, regardless of the nature of their roots—whether they are real, repeated, or complex.

Complex Numbers

When solving quadratic equations, you might encounter solutions that are not real numbers. These solutions are known as complex numbers. A complex number is expressed in the form \(a + bi\), where:

• \(a\) is the real part.
• \(b\) is the imaginary part, and \(i\) is the imaginary unit, defined as \(\sqrt{-1}\).

Complex numbers become necessary when the discriminant \(b^2 - 4ac\) in the quadratic formula is negative. A negative discriminant means the square root of a negative number is involved, leading to the presence of \(i\). Understanding complex numbers helps in fully capturing the scope of solutions possible for quadratic equations, far beyond just real numbers.

Discriminant

The discriminant is a vital component of the quadratic formula, represented by \(b^2 - 4ac\). It helps determine the nature of the roots of the quadratic equation. Here's how it works:

• If the discriminant is positive, there are two distinct real solutions.
• If it's zero, there is exactly one real solution, known as a repeated or double root.
• If the discriminant is negative, like in the problem above where it is \(-11\), the equation has two complex solutions.

By assessing the discriminant before solving with the quadratic formula, you can predict the type of solutions and approach the calculation with prior knowledge of what to expect.

Solving Equations

Solving quadratic equations requires understanding and applying various techniques and knowing when to use each. In this context:

• **Clear Fractions**: Begin with eliminating any fraction by multiplying through by the denominator, as seen in the original exercise.
• **Rearrange to Standard Form**: Ensure the equation is in the form \(ax^2 + bx + c = 0\) before applying any solving methods.
• **Use the Quadratic Formula**: This method is reliable for any quadratic equation. You've applied it by substituting the identified coefficients into the formula.

By systematically following these steps, you can efficiently solve any quadratic equation and confidently find its solutions, whether they are real or complex.