The Discriminant

Front

What is the discriminant in a quadratic equation?

Back

The discriminant is a value calculated from the coefficients of a quadratic equation in the form ax² + bx + c, given by the formula D = b² - 4ac. It helps determine the nature of the roots of the equation.

Front

How do you calculate the discriminant for the equation -2x² - x - 1 = 0?

Back

For the equation -2x² - x - 1, the coefficients are a = -2, b = -1, and c = -1. The discriminant is calculated as D = (-1)² - 4(-2)(-1) = 1 - 8 = -7.

Front

What does a discriminant of 0 indicate about the roots of a quadratic equation?

Back

A discriminant of 0 indicates that the quadratic equation has exactly one real solution, also known as a repeated or double root.

Front

If the discriminant is positive, what can be said about the roots of the quadratic equation?

Back

If the discriminant is positive, the quadratic equation has two distinct real solutions.

Front

What does a negative discriminant imply about the solutions of a quadratic equation?

Back

A negative discriminant implies that the quadratic equation has no real solutions, but two complex solutions.

Front

For the quadratic equation y = x² + 4x + 4, what is the discriminant and how many solutions does it have?

Back

The discriminant is 0, indicating that the equation has one real solution.

Front

How do you determine if the discriminant is positive, negative, or zero from a graph of a quadratic function?

Back

You can determine the nature of the discriminant by observing the number of x-intercepts (real roots) on the graph: two x-intercepts indicate a positive discriminant, one x-intercept indicates a zero discriminant, and no x-intercepts indicate a negative discriminant.

Front

What is the discriminant of the equation 6x² - 2x - 3 = 0?

Back

The discriminant is calculated as D = (-2)² - 4(6)(-3) = 4 + 72 = 76.

Front

What is the significance of the discriminant in solving quadratic equations?

Back

The discriminant helps predict the number and type of solutions (real or complex) for a quadratic equation without actually solving it.

Front

For the equation -8x² + 6x - 4 = 0, what is the discriminant and how many solutions does it have?

Back

The discriminant is -92, indicating that the equation has no real solutions.