Problem 59 - Elementary Algebra

Question

\(x^{2}-3 x+3=0\)

Step-by-Step Solution

Answer

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

Step 1 - Identify the coefficients

In the quadratic equation, identify the values of the coefficients. The standard form of a quadratic equation is x^2 + bx + c = 0. For the given equation, we have: \(a = 1\), \(b = -3\), and \(c = 3\).

Step 2 - Calculate the discriminant

The discriminant of a quadratic equation \(ax^2 + bx + c = 0\) is given by \(D = b^2 - 4ac\). Substitute the values of \(a\), \(b\), and \(c\) into this formula.\[ D = (-3)^2 - 4(1)(3) = 9 - 12 = -3\].

Step 3 - Analyze the discriminant

Since the discriminant \(D\) is less than 0 (\(D = -3\)), the quadratic equation \(x^2 - 3x + 3 = 0\) has no real roots. Instead, it has two complex conjugate roots.

Step 4 - Calculate the complex roots

Use the quadratic formula \(x = \frac{-b \pm \sqrt{D}}{2a}\) to find the complex roots. Here, substitute \(a = 1\), \(b = -3\), and \(D = -3\):\[x = \frac{-(-3) \pm \sqrt{-3}}{2(1)} = \frac{3 \pm \sqrt{3}i}{2}\]. Therefore, the roots are \(x = \frac{3 + \sqrt{3}i}{2}\) and \(x = \frac{3 - \sqrt{3}i}{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is an essential tool for solving quadratic equations. A quadratic equation is any equation that can be written in the standard form:
\[ax^2 + bx + c = 0\]
Here, \(a\), \(b\), and \(c\) are constants, and \(x\) represents the variable we want to solve for. The quadratic formula is given by:
\[x = \frac{-b \pm \sqrt{D}}{2a}\]
where \(D = b^2 - 4ac\) is the discriminant of the quadratic equation. Using this formula, you can find the solutions for \(x\) by substituting the coefficients \(a\), \(b\), and \(c\) from your specific equation. If you follow the formula's steps accurately, you will solve any quadratic equation, whether it has real or complex roots.

Discriminant

The discriminant \(D\) of a quadratic equation helps determine the nature of its roots. It is calculated using the formula:
\[D = b^2 - 4ac\]
Here's what the value of the discriminant tells us:

• If \(D > 0\), the equation has two distinct real roots.
• If \(D = 0\), the equation has exactly one real root (a repeated root).
• If \(D < 0\), the equation has two complex roots.

In the exercise, the discriminant is calculated as:
\[D = (-3)^2 - 4(1)(3) = 9 - 12 = -3\]
Since \(D < 0\), this tells us that the quadratic equation \(x^2 - 3x + 3 = 0\) has two complex roots.

Complex Roots

When the discriminant \(D\) of a quadratic equation is negative, the equation has complex roots. Complex roots appear in conjugate pairs (same real part but opposite imaginary parts). The quadratic formula to find these roots is still the same:
\[x = \frac{-b \pm \sqrt{D}}{2a}\]
However, since \(D < 0\), we write \(D\) as a negative number inside the square root. This introduces an imaginary unit \(i\), where \(i = \sqrt{-1}\). For the given exercise:
\[D = -3\]
So, replacing \(D\) with \(-3\) and solving:
\[x = \frac{-(-3) \pm \sqrt{-3}}{2(1)} = \frac{3 \pm \sqrt{3}i}{2}\]
Thus, the roots are \(x = \frac{3 + \sqrt{3}i}{2}\) and \(x = \frac{3 - \sqrt{3}i}{2}\). These roots have the same real part (\(\frac{3}{2}\)) but different imaginary parts (\(\frac{\sqrt{3}}{2}i\) and \(-\frac{\sqrt{3}}{2}i\)).

Quadratic Equation

A quadratic equation is any polynomial equation of the second degree, meaning the highest power of the variable \(x\) in the equation is \(x^2\). The general form of a quadratic equation is:
\[ax^2 + bx + c = 0\]
Where \(a\), \(b\), and \(c\) are coefficients, and \(x\) is the unknown variable. Here are a few key points:

• \(a\) should not be zero, as it would then no longer be a quadratic equation.
• The term \(ax^2\) ensures the 'quadratic' characteristic.
• The solutions to a quadratic equation can be found using factoring, completing the square, or the quadratic formula.

In the provided exercise, our quadratic equation is \(x^2 - 3x + 3 = 0\). By identifying the coefficients, \(a = 1\), \(b = -3\), and \(c = 3\), we use the quadratic formula to find the roots, handling real or complex numbers as needed. This makes solving quadratic equations systematic and straightforward.