Problem 58 - Algebra and Trigonometry

Question

Find all solutions of the equation and express them in the form \(a+b i .\) $$ x^{2}-3 x+3=0 $$

Step-by-Step Solution

Answer

The solutions are \(\frac{3}{2} + \frac{i\sqrt{3}}{2}\) and \(\frac{3}{2} - \frac{i\sqrt{3}}{2}\).

Step 1: Identify the Quadratic Equation

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

Step 2: Apply the Quadratic Formula

Use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the roots. Substitute the values of \(a\), \(b\), and \(c\) into the formula.

Step 3: Calculate the Discriminant

First, calculate the discriminant using \(b^2 - 4ac\).\[b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 3 = 9 - 12 = -3\].

Step 4: Solve for the Roots

Since the discriminant is negative, the roots will be complex numbers. Substitute \(b = -3\), \(a = 1\), and \(\text{Discriminant} = -3\) into the quadratic formula:\[x = \frac{-(-3) \pm \sqrt{-3}}{2 \cdot 1} = \frac{3 \pm i\sqrt{3}}{2}\].

Step 5: Express in the Form \(a + bi\)

The roots of the equation are in the form of \(a + bi\). Thus, the solutions are:\[x_1 = \frac{3}{2} + \frac{i\sqrt{3}}{2}\] and \[x_2 = \frac{3}{2} - \frac{i\sqrt{3}}{2}\].

Key Concepts

Complex Numbers

In the context of quadratic equations, complex numbers often arise when the discriminant (which we will talk about later) of the equation is negative. Complex numbers have two components: a real part and an imaginary part. The real part is the number without the imaginary unit, while the imaginary part includes the imaginary unit "i", which is defined by the property \(i^2 = -1\).

For instance, in the solutions given \(x_1 = \frac{3}{2} + \frac{i\sqrt{3}}{2}\) and \(x_2 = \frac{3}{2} - \frac{i\sqrt{3}}{2}\), \(\frac{3}{2}\) is the real part and \(\frac{i\sqrt{3}}{2}\) is the imaginary part. Complex numbers are often expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part "i".

• The complex number \(x_1\) can thus be seen as combination of \(\frac{3}{2}\) (the real) and \(\frac{\sqrt{3}}{2} i\) (the imaginary).
• Similarly, \(x_2\) combines \(\frac{3}{2}\) and \(-\frac{\sqrt{3}}{2} i\).

These components are crucial for solutions to polynomial equations when they don't cross the x-axis in the real number plane. Understanding that complex solutions arise from negative discriminants helps us solve more complex quadratic equations effectively.

Discriminant

The discriminant is a critical component of the quadratic formula, providing insight into the nature of the quadratic equation's roots. It is represented by the expression \(b^2 - 4ac\), derived from the quadratic equation \(ax^2 + bx + c = 0\).

• If the discriminant is positive, the quadratic equation has two distinct real roots.
• If the discriminant equals zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, as in our example where \(b^2 - 4ac = -3\), the equation has two complex conjugate roots.

In essence, the discriminant tells you whether the solutions to a quadratic equation are real or complex. For equations with a negative discriminant, it indicates the presence of complex numbers, implying that there is no intersection of the curve with the x-axis on a real number graph. This knowledge enables one to anticipate and solve for complex roots efficiently, often by expressing the roots in the form \(a + bi\).

Quadratic Formula

The quadratic formula is a mathematical tool used to find the solutions, or roots, of a quadratic equation. The general form of a quadratic equation is given by \(ax^2 + bx + c = 0\), and the quadratic formula can be expressed as:

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

This formula allows us to calculate the roots directly by substituting the values of \(a\), \(b\), and \(c\) from the quadratic equation. The plus-minus sign (\(\pm\)) indicates that there are usually two solutions for \(x\), which correspond to the two roots of the quadratic equation.

• In applications where the discriminant \(b^2 - 4ac\) is positive, both roots are real and distinct.
• When the discriminant is zero, the roots are real and identical.
• In cases where the discriminant is negative, as in our provided example, the roots are complex.

These roots, calculated through this formula, can then be expressed in the standard complex form, \(a + bi\). This powerful formula not only provides the roots of quadratic equations but also lends insight into the nature of these roots based on the sign of the discriminant.