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 of a quadratic equation ax² + bx + c = 0.

Front

What does the discriminant (b² - 4ac) indicate?

Back

The discriminant indicates the nature of the roots of a quadratic equation: if > 0, there are 2 real solutions; if = 0, there is 1 real solution; if < 0, there are 2 complex solutions.

Front

What are complex solutions?

Back

Complex solutions are solutions to equations that include imaginary numbers, typically in the form a + bi, where i is the imaginary unit.

Front

How do you find the complex solutions of 3x² + 2 = 0?

Back

Rearranging gives x² = -2/3. Taking the square root gives x = ±√(-2/3) = ±(√2/√3)i = ±(√6/3)i.

Front

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

Back

The zeros are x = ±8i, since x² = -64 leads to x = ±√(-64) = ±8i.

Front

If a quadratic has 2 real solutions, how many x-intercepts does it have?

Back

It has 2 x-intercepts.

Front

If a quadratic has 2 imaginary solutions, how many x-intercepts does it have?

Back

It has 0 x-intercepts.

Front

What is the significance of the vertex in a quadratic function?

Back

The vertex is the highest or lowest point of the parabola, representing the maximum or minimum value of the function.

Front

How do you determine if a quadratic opens upwards or downwards?

Back

If the coefficient of x² (a) is positive, the parabola opens upwards; if negative, it opens downwards.

Front

What is the axis of symmetry in a quadratic function?

Back

The axis of symmetry is the vertical line x = -b/(2a) that divides the parabola into two mirror-image halves.