The Discriminant

Front

What is the discriminant in a quadratic equation?

Back

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

Front

How does the value of the discriminant affect the number of real solutions of a quadratic equation?

Back

The value of the discriminant determines the nature of the roots: 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 no real solutions.

Front

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

Back

A positive discriminant indicates that the quadratic equation has 2 distinct real solutions.

Front

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

Back

A discriminant of zero indicates that the quadratic equation has exactly 1 real solution, which is a repeated root.

Front

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

Back

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

Front

How can you calculate the discriminant for the quadratic equation 2x² + 3x - 5?

Back

For the equation 2x² + 3x - 5, the discriminant is calculated as b² - 4ac = (3)² - 4(2)(-5) = 9 + 40 = 49.

Front

If the discriminant of a quadratic equation is -63, what can be concluded about its solutions?

Back

If the discriminant is -63, it indicates that the quadratic equation has no real solutions, only complex solutions.

Front

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

Back

The discriminant is derived from the coefficients of the quadratic equation ax² + bx + c, specifically using the formula b² - 4ac.

Front

How can the discriminant be used to determine the nature of the roots without solving the equation?

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By evaluating the discriminant, one can determine the nature of the roots: positive for two real solutions, zero for one real solution, and negative for no real solutions.

Front

What is the significance of the discriminant in real-world applications?

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The discriminant helps in determining the feasibility of solutions in various real-world problems modeled by quadratic equations, such as projectile motion and optimization problems.