Complex Numbers

Front

What are complex numbers?

Back

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit, defined as the square root of -1.

Front

What is the quadratic formula?

Back

The quadratic formula is used to find the solutions of a quadratic equation ax² + bx + c = 0 and is given by x = (-b ± √(b² - 4ac)) / (2a).

Front

What does the discriminant of a quadratic equation indicate?

Back

The discriminant, given by b² - 4ac, indicates the nature of the roots of the quadratic equation: if positive, there are two distinct real solutions; if zero, there is one real solution; if negative, there are two complex solutions.

Front

How do you calculate the sum of two complex numbers?

Back

To add two complex numbers, (a + bi) and (c + di), you add their real parts and their imaginary parts separately: (a + c) + (b + d)i.

Front

What is the significance of a discriminant of 0 in a quadratic equation?

Back

A discriminant of 0 indicates that the quadratic equation has exactly one real solution, meaning the graph of the equation touches the x-axis at one point (a double root).

Front

What is the imaginary unit 'i'?

Back

The imaginary unit 'i' is defined as the square root of -1, and it is used to express complex numbers.

Front

How can you determine if a quadratic equation has complex solutions?

Back

A quadratic equation has complex solutions if its discriminant (b² - 4ac) is negative.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Front

What is the relationship between the roots and coefficients of a quadratic equation?

Back

For a quadratic equation ax² + bx + c = 0, the sum of the roots is -b/a and the product of the roots is c/a.

Front

How do you find the x-intercepts of a quadratic function?

Back

To find the x-intercepts of a quadratic function, set the function equal to zero and solve for x using the quadratic formula or by factoring.