Problem 66 - Precalculus Mathematics for Calculus

Question

Find all solutions of the equation and express them in the form \(a+b i\) $$2 x^{2}+3=2 x$$

Step-by-Step Solution

Answer

Solutions are \(\frac{1}{2} \pm \frac{\sqrt{5}}{2} i\).

Step 1: Rewrite the Equation

The given equation is \(2x^2 + 3 = 2x\). We need to rewrite the equation in standard quadratic form. To do this, subtract \(2x\) from both sides: \(2x^2 - 2x + 3 = 0\). Now, we have a quadratic equation in the form \(ax^2 + bx + c = 0\) where \(a = 2\), \(b = -2\), and \(c = 3\).

Step 2: Identify Coefficients

Identify the coefficients from the equation \(2x^2 - 2x + 3 = 0\). These coefficients are important for using the quadratic formula later. Here, \(a = 2\), \(b = -2\), and \(c = 3\).

Step 3: Use the Quadratic Formula

The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Substitute the values of \(a\), \(b\), and \(c\) into the formula: \[x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}\].

Step 4: Evaluate Discriminant

Calculate the discriminant \(b^2 - 4ac\), which is the part under the square root sign in the quadratic formula. Here, it is \((-2)^2 - 4 \cdot 2 \cdot 3 = 4 - 24 = -20\). Since the discriminant is negative, the solutions will be complex.

Step 5: Solve for Roots

Using the quadratic formula and the discriminant, compute the roots: \[x = \frac{2 \pm \sqrt{-20}}{4}\]. Simplify further to express \(\sqrt{-20}\) as \(2i\sqrt{5}\): \[x = \frac{2 \pm 2i \sqrt{5}}{4} = \frac{1 \pm i \sqrt{5}}{2}\].

Step 6: Express Solutions in Form \(a+bi\)

The solutions can be written in the standard complex number form \(a + bi\). Hence, the solutions are \(\frac{1}{2} + \frac{\sqrt{5}}{2} i\) and \(\frac{1}{2} - \frac{\sqrt{5}}{2} i\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are vital in mathematics, and they often appear in the form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, with \(a eq 0\) since the equation would otherwise not be quadratic. These equations can describe a wide range of phenomena, from the arc of a baseball to complex natural calculations in physics and engineering.

To solve quadratic equations, several methods can be employed:

• Factoring
• Completing the square
• Using the quadratic formula

In our original exercise, the equation \(2x^2 - 2x + 3 = 0\) is already in the necessary form after a little rearranging.

This prepares us for the next steps, especially identifying the coefficients needed for further mathematical computations.

Demystifying the Discriminant

The discriminant is a crucial part of solving quadratic equations through the quadratic formula. It is represented as \(b^2 - 4ac\) where \(a\), \(b\), and \(c\) are coefficients from \(ax^2 + bx + c = 0\). The discriminant determines the nature and number of solutions:

• If \(b^2 - 4ac > 0\), the equation has two distinct real roots.
• If \(b^2 - 4ac = 0\), there is exactly one real root, also known as a repeated root.
• If \(b^2 - 4ac < 0\), the equation has two complex roots.

In the problem we tackled, the discriminant was computed to be \(-20\).

This negative value suggests that the roots are complex numbers, which feature imaginary components. This knowledge paves the way for employing the quadratic formula to find these complex solutions.

Using the Quadratic Formula

The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a guaranteed method for solving quadratic equations.

With it, given the coefficients \(a\), \(b\), and \(c\), you can compute the roots, including complex ones when the discriminant is negative. Applying this to our equation, the values of \(a = 2\), \(b = -2\), and \(c = 3\) were substituted into the formula:

• Calculate \(-b\) which equals \(2\)
• Compute the discriminant \(\sqrt{-20}\), simplifying to \(2i\sqrt{5}\)
• The results are the complex roots: \(\frac{1 \pm i \sqrt{5}}{2}\)

This solution presents the roots in the form \(a + bi\), helping us understand their interaction as complex numbers. The technique not only allows finding the roots but also interpreting their real and imaginary parts, which is essential in the broader understanding of complex numbers.