3.2/ 3.3 Complex Numbers and Complete the Square 24-25 H

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 does it mean to complete the square?

Back

Completing the square is a method used to convert a quadratic equation of the form ax^2 + bx + c into the form (x - p)^2 = q, making it easier to solve.

Front

How do you complete the square for the equation x^2 + 6x - 3 = 0?

Back

1. Move -3 to the other side: x^2 + 6x = 3. 2. Take half of 6 (which is 3), square it (9), and add to both sides: (x + 3)^2 = 12. 3. Solve for x: x = -3 ± √12.

Front

What is the vertex form of a quadratic equation?

Back

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

Front

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

Back

The discriminant (D = b^2 - 4ac) determines the nature of the roots of the quadratic equation: D > 0 means two distinct real roots, D = 0 means one real root, and D < 0 means two complex roots.

Front

How do you solve the equation (x - 3)^2 = -20?

Back

1. Recognize that the equation has no real solutions since the square of a real number cannot be negative. 2. Rewrite as (x - 3)^2 = 20i^2. 3. Solve for x: x = 3 ± 2√5i.

Front

What is the formula for the roots of a quadratic equation?

Back

The roots of a quadratic equation ax^2 + bx + c = 0 can be found using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

Front

What is the process of converting a quadratic equation to vertex form?

Back

To convert to vertex form, complete the square on the quadratic expression. For example, for x^2 - 4x + 4, rewrite as (x - 2)^2.

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as i = √(-1). It is used to express complex numbers.

Front

How do you identify the vertex of a parabola from its equation?

Back

For a quadratic in standard form y = ax^2 + bx + c, the vertex can be found at the point (h, k) where h = -b/(2a) and k = f(h).