91Ó°ÊÓ

Quadratic equations are polynomial equations of degree 2, written in the general form ax^2 + bx + c = 0. They are called 'quadratic' because the highest exponent of the variable is 2. Quadratic equations can have different types of solutions: real and distinct, real and repeated, or complex. To solve a quadratic equation, one can use various methods:

• Factoring
• Completing the square
• Using the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

In our specific problem, the quadratic equation is in the form x^2 = 100. This is a simpler case because there is no middle term (bx) or constant term (c), making it an ideal example for taking square roots directly.

The square root of a number x is a value y such that y^2 = x. It is important to remember that every positive real number has two square roots: one positive and one negative. For example, both 10 and -10 are square roots of 100 because (10)^2 = 100 and (-10)^2 = 100.
When solving equations involving square roots:

• Always consider both the positive and negative roots.
• Use the notation \( \pm \sqrt{x} \) to represent both roots.

In our exercise, after taking the square root of both sides of the equation x^2 = 100, we get x = \pm \sqrt{100}, which simplifies to x = \pm 10.

The complex plane is a two-dimensional plane for representing complex numbers. Each complex number is written as a + bi, where a is the real part and bi is the imaginary part. The complex plane is similar to the Cartesian plane used in graphing:

• The horizontal axis (x-axis) represents the real part.
• The vertical axis (y-axis) represents the imaginary part.

For our equation x^2 = 100, we were looking for complex-number solutions. Even though 10 and -10 are real numbers, they can still be represented in the complex plane as 10 + 0i and -10 + 0i, respectively. This emphasizes that every real number is also a complex number with an imaginary part of 0.
Complex solutions become more visible in cases where the solutions are not purely real. For instance, solving x^2 = -1 results in the complex numbers i and -i, where i is the imaginary unit (\( i^2 = -1 \)).