Problem 38 - Algebra and Trigonometry

Question

\(31-42\) . Solve the equation both algebraically and graphically. $$ x^{2}+3=2 x $$

Step-by-Step Solution

Answer

There are no real solutions; the complex solutions are \(x = 1 \pm i\sqrt{2}\).

Step 1: Organize the Equation

The given equation is \(x^2 + 3 = 2x\). First, we will rearrange all terms to one side of the equation to set it to 0. Subtract \(2x\) from both sides, giving us \(x^2 - 2x + 3 = 0\).

Step 2: Solve Algebraically using the Quadratic Formula

To solve \(x^2 - 2x + 3 = 0\), we use the quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = -2\), and \(c = 3\). Calculate the discriminant: \(b^2 - 4ac = (-2)^2 - 4\cdot1\cdot3 = 4 - 12 = -8\). Since the discriminant is negative, there are no real solutions, only complex solutions.

Step 3: Identify Complex Solutions

Since the discriminant is negative, the solutions are complex. Substitute into the formula: \(x = \frac{-(-2) \pm \sqrt{-8}}{2}\). Simplifying gives \(x = \frac{2 \pm 2i\sqrt{2}}{2}\), resulting in \(x = 1 \pm i\sqrt{2}\).

Step 4: Graphical Representation

To visualize the solutions, consider the graph of \(y = x^2 - 2x + 3\). Since this is a parabola opening upwards with a positive \(a\) value (\(1\)), and the vertex is above the x-axis, it doesn't intersect the x-axis. No real roots means the solutions are indeed complex.

Key Concepts

Complex Numbers

Complex numbers appear when solving quadratic equations with a negative discriminant. A complex number consists of a real part and an imaginary part, represented as \(a + bi\). In this example, the solutions for the quadratic equation are \(x = 1 \pm i\sqrt{2}\), where:

• the real part is 1, and
• the imaginary part is \(i\sqrt{2}\).

In complex numbers, \(i\) is the imaginary unit, defined as \(i = \sqrt{-1}\). This means \(i^2 = -1\). Complex numbers can be visualized on the complex plane, with the real part on the x-axis and the imaginary part on the y-axis. This makes understanding how they relate to equations much easier. Many real-world applications, such as electrical engineering and physics, use complex numbers to describe oscillating systems or circuits.

Quadratic Formula

The quadratic formula is a powerful tool used to find the roots of any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]To effectively use this formula, remember to correctly identify the coefficients \(a\), \(b\), and \(c\). In our exercise:

• \(a = 1\)
• \(b = -2\)
• \(c = 3\)

The quadratic formula simplifies the process of solving equations by considering the potential real and complex roots. Calculating the expression under the square root, known as the discriminant, helps determine the nature of the roots. It is an essential formula in algebra, allowing you to move from real to complex analysis when necessary.

Discriminant

The discriminant in the quadratic formula is the part under the square root, \(b^2 - 4ac\). It tells us whether the quadratic equation has real or complex solutions:

• If the discriminant is positive, there are two real and distinct solutions.
• If it is zero, there is one real, repeated solution (or a double root).
• If it is negative, like in our problem where \(b^2 - 4ac = -8\), the solutions are complex.

A negative discriminant indicates no x-intercepts on the graph, signifying complex solutions. These solutions are conjugates, meaning they share the same real part and have equal but opposite imaginary parts, as seen in \(x = 1 \pm i\sqrt{2}\). Understanding the discriminant aids in graphing and solving quadratic equations more intuitively.