Problem 24 - Algebra A Combined Function

Question

Use the quadratic formula to solve each equation. These equations have real solutions and complex, but not real, solutions. $$ x^{2}+2 x+2=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -1 + i\) and \(x = -1 - i\).

Step 1: Identify Coefficients

First, identify the coefficients for the quadratic equation, which is in the form \(ax^2 + bx + c = 0\). Here, \(a = 1\), \(b = 2\), and \(c = 2\).

Step 2: Write the Quadratic Formula

The quadratic formula is used to find the roots of a quadratic equation. It is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Calculate the Discriminant

The discriminant \(b^2 - 4ac\) determines the nature of the roots. Substitute the values: \[ b^2 - 4ac = 2^2 - 4 imes 1 imes 2 = 4 - 8 = -4 \]

Step 4: Discriminant Indicates Complex Solutions

Since the discriminant \(-4\) is negative, the quadratic equation has complex solutions.

Step 5: Apply the Quadratic Formula

Substitute the values of \(a\), \(b\), and \(c\) into the quadratic formula: \[ x = \frac{-2 \pm \sqrt{-4}}{2 \times 1} \] Simplify under the square root, \(\sqrt{-4} = 2i\), where \(i\) is the imaginary unit.

Step 6: Simplify the Expression

Now, simplify the equation to solve for \(x\): \[ x = \frac{-2 \pm 2i}{2} \]Split this fraction: \[ x = -1 \pm i \]

Key Concepts

Quadratic Equation Solutions

Quadratic equations take the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. These equations can have one, two, or no real solutions, all depending on the discriminant which we'll discuss later. The solutions to quadratic equations are the x-values, or roots, where the equation is satisfied, meaning it equals zero. To find these solutions, the quadratic formula is often used, especially when the quadratic does not easily factor. The formula is given by:

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

The symbol \(\pm\) in the formula indicates two possible solutions: one adding the square root and one subtracting it. By using this formula, we can find the roots of any quadratic equation, whether the solutions are real or complex. It's a go-to method when other techniques, like factoring, aren't feasible.

Discriminant in Quadratic Equations

The discriminant, represented by \(b^2 - 4ac\), plays a critical role in helping us understand the nature of the roots in quadratic equations. It is derived from within the square root part of the quadratic formula. Depending on its value, we can determine what types of solutions our equation has. Here’s what you need to know:

• If \(b^2 - 4ac > 0\), the equation has two distinct real solutions.
• If \(b^2 - 4ac = 0\), there is one real solution, sometimes called a repeated or double root.
• If \(b^2 - 4ac < 0\), the roots are complex, indicating that they are not real numbers.

In our specific example, the discriminant was calculated to be \(-4\). Since this value is negative, the equation has complex roots. This means that while the solutions are not real numbers, they exist within the realm of complex numbers, which have both a real and an imaginary part.

Complex Numbers in Algebra

When faced with a negative discriminant in a quadratic equation, complex numbers come into play. Complex numbers extend the idea of one-dimensional numbers (like real numbers) to a two-dimensional plane, allowing them to express solutions involving the square root of negative numbers. These are written in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In our example, \(\sqrt{-4}\) becomes \(2i\), where \(i\) represents the imaginary unit, defined as \(\sqrt{-1}\).

• The real part of complex numbers often involves coefficients from the original equation.
• The imaginary part results directly from the negative discriminant.

Thus, the roots become \(-1 \pm i\), showing both a real and imaginary component. Complex numbers are crucial in various fields, including engineering and physics, looking beyond the limitations imposed by real numbers.

Solving Quadratic Equations

To solve any quadratic equation, you can use various methods, like factoring, completing the square, or the quadratic formula. In scenarios where the equation is not factorable or the solutions are complex, the quadratic formula provides a reliable solution. Here's a quick guide to effectively use the formula:

• Identify coefficients: Spot the values of \(a\), \(b\), and \(c\) in the equation.
• Calculate the discriminant: Work out \(b^2 - 4ac\) to determine the nature of the roots.
• Substitute in the formula: Plug \(a\), \(b\), and \(c\) into the quadratic formula.
• Simplify: Solve the resulting expression to find the roots.

With these steps, you'll be equipped to tackle any quadratic equation. Remember, practice is key to mastering this mathematical process. By understanding each component thoroughly, you'll simplify complex equations with confidence.