Quadratic Formula with Complex Roots Flashcard

Front

What does the discriminant of a quadratic equation indicate about the nature of its roots?

Back

The discriminant indicates the nature of the roots: if it is positive, there are two distinct real roots; if it is zero, there is one real repeated root; if it is negative, there are two complex (imaginary) roots.

Front

What is the quadratic formula used to find the roots of a quadratic equation?

Back

The quadratic formula is given by x = (-b ± √(b² - 4ac)) / (2a), where a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0.

Front

What is the value of the discriminant for the equation 2x² + 4x + 2 = 0?

Back

The discriminant is calculated as b² - 4ac = 4² - 4(2)(2) = 16 - 16 = 0, indicating one real repeated solution.

Front

How do you simplify the square root of a negative number, such as √-16?

Back

To simplify √-16, you can express it as √(16) * √(-1) = 4i, where i is the imaginary unit.

Front

What are complex roots in the context of quadratic equations?

Back

Complex roots are solutions to quadratic equations that involve imaginary numbers, occurring when the discriminant is negative.

Front

What is the significance of the imaginary unit 'i' in mathematics?

Back

The imaginary unit 'i' is defined as √(-1) and is used to express complex numbers, which have a real part and an imaginary part.

Front

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

Back

A quadratic equation has complex roots if the discriminant (b² - 4ac) is less than zero.

Front

What is the simplified form of √-49?

Back

The simplified form of √-49 is 7i.

Front

What does it mean for a quadratic equation to have one real repeated solution?

Back

It means that the quadratic equation has exactly one solution, which occurs when the discriminant is zero, resulting in a double root.

Front

How do you apply the quadratic formula to solve the equation -2x² - x - 7 = 0?

Back

To apply the quadratic formula, identify a = -2, b = -1, c = -7, then calculate the discriminant and substitute into the formula to find the roots.