Problem 46 - Precalculus: Functions and Graphs

Question

Find the solutions of the equation. $$-3 x^{2}+x-5=0$$

Step-by-Step Solution

Answer

The equation has no real solutions; its complex solutions are \(x = \frac{1}{6} \pm \frac{i\sqrt{59}}{6}\).

Step 1: Identify the Quadratic Equation

The given equation is a quadratic equation in the standard form \(ax^2 + bx + c = 0\), where \(a = -3\), \(b = 1\), and \(c = -5\).

Step 2: Apply the Quadratic Formula

The solutions to the quadratic equation \(ax^2 + bx + c = 0\) can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \(a = -3\), \(b = 1\), and \(c = -5\) into the formula.

Step 3: Calculate the Discriminant

Calculate the discriminant using \(b^2 - 4ac\). For this equation:\[ 1^2 - 4(-3)(-5) = 1 - 60 = -59 \]The discriminant is \(-59\).

Step 4: Interpret the Discriminant

Since the discriminant is negative, the quadratic equation has no real solutions. Instead, it has two complex solutions.

Step 5: Find the Complex Solutions

Using the quadratic formula, find the complex solutions:\[ x = \frac{-1 \pm \sqrt{-59}}{-6} \]This can be rewritten as:\[ x = \frac{-1}{-6} \pm \frac{i\sqrt{59}}{6} \]Simplify to get:\[ x = \frac{1}{6} \pm \frac{i\sqrt{59}}{6} \]

Key Concepts

Complex Solutions

When working with quadratic equations, sometimes you encounter a situation where the solutions aren't real numbers but instead are complex numbers. This usually happens when the discriminant, which we will discuss in detail later, is negative.

Complex solutions mean that our solutions involve imaginary numbers. Imaginary numbers are expressed with the letter \(i\), where \(i = \sqrt{-1}\). When you have a negative discriminant, the square root of this negative number brings in the imaginary unit \(i\).

• This leads to solutions of the form \(a + bi\) or \(a - bi\), where \(a\) and \(b\) are real numbers.
• For our equation, the complex solutions are \( x = \frac{1}{6} \pm \frac{i \sqrt{59}}{6} \).
• The real part here is \(\frac{1}{6}\), and the imaginary part is \(\frac{i \sqrt{59}}{6}\).

Complex solutions are essential when you're solving equations in fields like engineering and physics, where both real and imaginary components can offer significant insights.

Quadratic Formula

The quadratic formula is like a powerhouse tool you can use to solve any quadratic equation, which is an equation in the form \(ax^2 + bx + c = 0\). It's particularly useful when factoring isn't straightforward or possible.

The quadratic formula itself is given by:

• \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \(b^2 - 4ac\) is called the discriminant, and it plays a crucial role in determining the type of solutions you'll get.

This formula works universally for all quadratic equations, delivering either real or complex solutions based on the discriminant's value. Here’s how you can apply it:

• Start by identifying the coefficients \(a\), \(b\), and \(c\) from your equation.
• Plug these values into the formula.
• Simplify to find the solutions, keeping an eye on the discriminant.

Remember, the "\(\pm\)" in the formula means you’ll often get two solutions—one with addition and the other with subtraction.

Discriminant in Quadratic Equations

The discriminant is the part of the quadratic formula under the square root, \(b^2 - 4ac\). It is crucial because it tells us the nature of the roots of the quadratic equation.

Here's how the discriminant helps:

• If \(b^2 - 4ac > 0\), you have two distinct real solutions.
• If \(b^2 - 4ac = 0\), you have exactly one real solution or a repeated real root.
• If \(b^2 - 4ac < 0\), you have two complex solutions (as in our exercise).

The discriminant allows you to anticipate the solution type just by calculating a single expression. In our specific equation, we calculated the discriminant as \(-59\), which is negative, leading us to expect complex solutions.

Understanding the discriminant's implications helps you choose efficient methods and anticipate what kind of numbers you're dealing with. This can enhance problem-solving techniques in both academic exercises and real-world applications.