Discriminant

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is given by the formula b² - 4ac, where a, b, and c are coefficients of the quadratic equation ax² + bx + c = 0.

Front

How does the discriminant determine the nature of the roots of a quadratic equation?

Back

If the discriminant is positive, there are 2 distinct real solutions. If it is zero, there is 1 real solution (a repeated root). If it is negative, there are 2 imaginary solutions.

Front

Calculate the discriminant for the quadratic equation 2x² + 4x + 2 = 0.

Back

The discriminant is 4 - 4(2)(2) = 4 - 16 = -12 (negative, so 2 imaginary solutions).

Front

What does a positive discriminant indicate about the graph of a quadratic function?

Back

A positive discriminant indicates that the graph of the quadratic function intersects the x-axis at two points, meaning there are two distinct real solutions.

Front

What does a zero discriminant indicate about the graph of a quadratic function?

Back

A zero discriminant indicates that the graph of the quadratic function touches the x-axis at one point, meaning there is one real solution (a repeated root).

Front

What does a negative discriminant indicate about the graph of a quadratic function?

Back

A negative discriminant indicates that the graph of the quadratic function does not intersect the x-axis, meaning there are two imaginary solutions.

Front

If the discriminant of a quadratic equation is b² - 4ac, what is the value of the discriminant for the equation x² + 6x + 9 = 0?

Back

The discriminant is 6² - 4(1)(9) = 36 - 36 = 0.

Front

For the quadratic equation 3x² - 12x + 9 = 0, what is the discriminant?

Back

The discriminant is (-12)² - 4(3)(9) = 144 - 108 = 36 (positive, so 2 distinct real solutions).

Front

What is the significance of the coefficients a, b, and c in the quadratic formula?

Back

In the quadratic equation ax² + bx + c = 0, 'a' determines the width and direction of the parabola, 'b' affects the position of the vertex, and 'c' is the y-intercept.

Front

How can you determine the number of solutions for the equation 4x² + 4x + 1 = 0 using the discriminant?

Back

Calculate the discriminant: 4² - 4(4)(1) = 16 - 16 = 0, indicating 1 real solution.