Alg. 2- Imaginary, Complex Numbers, & Completing the Square

Front

What is a complex number?

Back

A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit, defined as the square root of -1.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as i = √(-1). It is used to extend the real number system to include solutions to equations that do not have real solutions.

Front

How do you complete the square for the equation x² + bx = c?

Back

To complete the square, take half of b, square it, and add it to both sides of the equation. The left side then factors into (x + b/2)².

Front

What is the vertex form of a quadratic equation?

Back

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

What does it mean to factor a quadratic expression?

Back

Factoring a quadratic expression means rewriting it as a product of two binomials, such as x² - 4x + 4 = (x - 2)².

Front

How do you add complex numbers?

Back

To add complex numbers, combine the real parts and the imaginary parts separately. For example, (a + bi) + (c + di) = (a + c) + (b + d)i.

Front

What is the product of two complex numbers (a + bi)(c + di)?

Back

The product is given by (ac - bd) + (ad + bc)i.

Front

What is the conjugate of a complex number a + bi?

Back

The conjugate of a complex number a + bi is a - bi.

Front

How do you simplify the expression -3i(4 + 2i)?

Back

Distributing gives -12i - 6i². Since i² = -1, this simplifies to 6 - 12i.

Front

What is the significance of the discriminant in a quadratic equation?

Back

The discriminant (b² - 4ac) determines the nature of the roots: if positive, there are two distinct real roots; if zero, one real root; if negative, two complex roots.