Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-14,\) use the quadratic formula to find the imaginary roots of each equation. $$ 4 x^{2}-4 x+5=0 $$

Step-by-Step Solution

Answer

The roots are \(x = \frac{1}{2} + i\) and \(x = \frac{1}{2} - i\).

Step 1: Identify Coefficients

For the quadratic equation in the form of \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). In the given equation \(4x^2 - 4x + 5 = 0\), we have \(a = 4\), \(b = -4\), and \(c = 5\).

Step 2: Write the Quadratic Formula

The quadratic formula to find the roots of the equation \(ax^2 + bx + c = 0\) is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Compute the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Substitute the values: \((-4)^2 - 4 \times 4 \times 5 = 16 - 80 = -64\). The discriminant is \(-64\), which is negative, indicating that the roots are imaginary.

Step 4: Substitute into the Quadratic Formula

Use the quadratic formula with the discriminant value. Substitute \(b = -4\), \(a = 4\), and the discriminant \(-64\) into the formula:\[x = \frac{-(-4) \pm \sqrt{-64}}{2 \times 4}\]This becomes:\[x = \frac{4 \pm \sqrt{-64}}{8}\]

Step 5: Simplify the Expression

Recognize \(\sqrt{-64}\) as \(8i\) because \(\sqrt{-64} = \sqrt{-1 \times 64} = 8i\) (where \(i\) is the imaginary unit). The expression becomes:\[x = \frac{4 \pm 8i}{8}\]

Step 6: Divide to Find the Roots

Simplify the fractions by dividing each term by \(8\):\[x = \frac{4}{8} \pm \frac{8i}{8}\]This reduces to:\[x = \frac{1}{2} \pm i\]

Step 7: Finalize the Roots

The imaginary roots of the equation are \(x = \frac{1}{2} + i\) and \(x = \frac{1}{2} - i\).

Key Concepts

Imaginary Roots

In mathematics, roots of an equation are the values that satisfy the equation when substituted for the variable. Roots can be real or imaginary. Imaginary roots occur when the quadratic equation has no real solutions. These are characterized by the presence of the imaginary unit, denoted as \(i\), which is defined as \(i = \sqrt{-1}\). When a quadratic equation features imaginary roots, it means the parabola does not intersect the x-axis in its graph.

Imaginary roots typically arise when the discriminant of the quadratic equation is negative. Most quadratic equations have two roots, and when these roots are imaginary, they appear as complex conjugates, such as \(a + bi\) and \(a - bi\), where \(a\) and \(b\) are real numbers.

Discriminant

The discriminant is a crucial part of the quadratic formula, helping us determine the nature of the roots. It's found within the quadratic formula under the square root symbol: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

The discriminant itself is the expression \(b^2 - 4ac\). Here’s how the value of the discriminant affects the roots of a quadratic equation:

• If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, or rather, two identical real roots.
• If \(b^2 - 4ac < 0\), the equation has two complex, or imaginary, roots.

In the solution we worked through, a negative discriminant indicated imaginary roots.

Complex Numbers

Complex numbers are an extension of the real numbers and include a real part and an imaginary part. They are expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part (\(b\) is a real number and \(i = \sqrt{-1}\)). Complex numbers are incredibly useful in various fields, including engineering and physics, because they offer a way to handle equations that have no real solutions.

These numbers adhere to standard rules of arithmetic with additional properties tailored to them:

• \(i^2 = -1\), which forms the basis of operations involving \(i\).
• Complex conjugates, like \(a + bi\) and \(a - bi\), have their product purely real: \((a+bi)(a-bi) = a^2 + b^2\).

When solving equations that yield complex numbers, understanding these principles helps simplify and find the required solutions.

Completing the Square

Completing the square is a technique used to solve quadratic equations and understand the properties of parabolas. The method involves reformatting a quadratic equation into a perfect square trinomial, thus making it easier to solve.

Given a standard quadratic equation, \(ax^2 + bx + c = 0\):

• Divide all terms by \(a\) if \(a eq 1\), to simplify (if necessary).
• Move the constant term to the other side of the equation.
• Take half of the coefficient of \(x\), square it, and add to both sides of the equation.
• The left-hand side now becomes a complete square \((x+d)^2\), aiding in solving for \(x\).

Completing the square is especially useful for understanding the vertex form of a parabola and provides deeper insight into its graph.