4.56: 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. It is given by the formula D = b² - 4ac and determines 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 = 0, 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 negative discriminant indicate about the solutions of a quadratic equation?

Back

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

Front

What is the significance of the numerator in a rational expression?

Back

The numerator in a rational expression represents the quantity that is being divided by the denominator. It can affect the value of the expression depending on its value.

Front

What is the denominator in a rational expression?

Back

The denominator in a rational expression is the quantity by which the numerator is divided. It cannot be zero, as division by zero is undefined.

Front

How do you simplify a rational expression?

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To simplify a rational expression, factor both the numerator and the denominator and then cancel any common factors.

Front

What is the formula for the discriminant of a quadratic equation?

Back

The formula for the discriminant of a quadratic equation ax² + bx + c is D = b² - 4ac.

Front

What does a discriminant of zero indicate about the solutions of a quadratic equation?

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

Front

How many solutions does a quadratic equation have if the discriminant is positive?

Back

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

Front

What is the relationship between the coefficients of a quadratic equation and its discriminant?

Back

The coefficients a, b, and c of a quadratic equation ax² + bx + c directly influence the value of the discriminant D = b² - 4ac, which determines the nature of the roots.