N.9 - Quadratic Formula with Complex Solutions

Front

What is the Quadratic Formula?

Back

The Quadratic Formula is x = (-b ± √(b² - 4ac)) / (2a), used to find the solutions (or zeros) of a quadratic equation ax² + bx + c = 0.

Front

What does it mean if the discriminant (b² - 4ac) is negative?

Back

If the discriminant is negative, the quadratic equation has two complex (imaginary) solutions.

Front

How many x-intercepts does a quadratic function have if it has one real solution?

Back

It has 1 x-intercept.

Front

What are the zeros of the function f(x) = x² + 64?

Back

The zeros are -8i and 8i.

Front

Solve the equation -2x² + 4x = 9.

Back

The solutions are (2 - i√14)/2 and (2 + i√14)/2.

Front

What is the significance of the term 'imaginary solutions'?

Back

Imaginary solutions occur when the discriminant is negative, indicating that the graph of the quadratic does not intersect the x-axis.

Front

If a quadratic has two distinct real solutions, what can be said about the discriminant?

Back

The discriminant (b² - 4ac) is positive.

Front

Solve the equation x² - 5x + 10 = 0.

Back

The solutions are (5 - i√15)/2 and (5 + i√15)/2.

Front

What is the relationship between the number of solutions and the discriminant?

Back

If the discriminant is positive, there are 2 real solutions; if it is zero, there is 1 real solution; if it is negative, there are 2 imaginary solutions.

Front

What does a real double solution mean in the context of quadratics?

Back

A real double solution occurs when the discriminant is zero, indicating the vertex of the parabola touches the x-axis.