Question

Solve the quadratic equation. $$x^{2}-2 x+2=0$$

Short answer

The solutions to the quadratic equation are \(x = 1 + i\) and \(x = 1 - i\)

Step-by-step solution

Identify a, b, and c

From the equation \(x^{2}-2x+2=0\), we can identify: \(a=1\),\(b=-2\),\(c=2\)

Apply the quadratic formula

We can set up the quadratic formula: \(\frac{-b \pm \sqrt{b^{2}-4ac}}{2a}\),Substituting the values where \(a=1\), \(b=-2\), and \(c=2\), we get: \(x = \frac{-(-2) \pm \sqrt{(-2)^{2}-4*1*2}}{2*1}\)

Solve the equation

Simplifying the above expression, we have: \(x = \frac{2 \pm \sqrt{4-8}}{2}\), It simplifies to: \(x = \frac{2 \pm \sqrt{-4}}{2}\) we can express \(\sqrt{-4}\) as \(2i\). So, \(x = \frac{2 \pm 2i}{2}\), which simplifies to: \(x = 1 \pm i\)

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are of the form \(ax^2 + bx + c = 0\). The formula is \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]. Using this formula, we can find the solutions (also called roots) of any quadratic equation by substituting in the values of \(a\), \(b\), and \(c\) from the equation.

For instance, in our exercise \(x^2-2x+2=0\), \(a=1\), \(b=-2\), and \(c=2\). The quadratic formula simplifies the process of solving quadratic equations by providing a direct method to calculate the roots, regardless of whether they are real or complex numbers.

Complex Numbers

Complex numbers extend the idea of the traditional number line to a two-dimensional plane, by including the 'imaginary' unit \(i\), where \(i^2 = -1\). A complex number is written as \(a + bi\), with \(a\) being the real part and \(b\) the imaginary part.

Role in Quadratic Equations

Complex numbers are particularly helpful when the discriminant in the quadratic formula (\(b^2-4ac\)) is negative, leading to imaginary solutions. As seen in our solved exercise, the square root of a negative number, such as \(\sqrt{-4}\), is expressed as \(2i\), with \(i\) indicating the presence of an imaginary component. This concept is crucial since it guarantees a solution to every quadratic equation, even when the solutions are not on the real number line.

Solving Quadratic Equations

Solving quadratic equations is about finding the values of \(x\) that make the equation true. There are several ways to solve these equations: factoring, completing the square, graphing, and using the quadratic formula. The latter is often the most versatile as it can handle a broader range of quadratics—those that do not factor easily, for instance.

In our example, after applying the quadratic formula and simplifying, we find that \(x = 1 \pm i\), which signifies we've found two complex solutions. This result implies that our quadratic equation does not intersect the x-axis, which graphically represents the absence of real solutions.

Discriminant of a Quadratic

The discriminant of a quadratic equation, symbolized as \(\Delta\) and calculated by the expression \(b^2-4ac\), plays a crucial role in determining the nature of the roots. The discriminant tells us whether the solutions are real or complex and whether they are distinct or repeated.

For example:

• If \(\Delta > 0\), the equation has two distinct real solutions.
• If \(\Delta = 0\), there is exactly one real solution (a repeated root).
• If \(\Delta < 0\), there are two complex solutions.

In the exercise, the discriminant is \(-4\), indicating the presence of complex solutions. Understanding the discriminant gives us a quick way to predict the nature of the equation's solutions without fully solving it.