4.6 Quadratic Formula and Discriminant

Front

What is the Quadratic Formula?

Back

The Quadratic Formula is used to find the solutions (roots) of a quadratic equation in the form ax² + bx + c = 0. It is given by: x = (-b ± √(b² - 4ac)) / (2a).

Front

What does the discriminant tell us about the roots of a quadratic equation?

Back

The discriminant (D = b² - 4ac) indicates the nature of the roots of a quadratic equation: If D > 0, there are 2 distinct real roots; If D = 0, there is 1 real root (a repeated root); If D < 0, there are 2 complex (imaginary) roots.

Front

Calculate the discriminant for the equation: x² + 5x + 6.

Back

D = b² - 4ac = 5² - 4(1)(6) = 25 - 24 = 1 (2 distinct real roots).

Front

What is the significance of a discriminant equal to zero?

Back

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

Front

Solve the quadratic equation using the formula: 3x² - 12x + 9 = 0.

Back

x = (12 ± √(0)) / 6 = 2 (one real solution).

Front

What are the roots of the equation: x² - 4 = 0?

Back

x = ±2 (two distinct real roots).

Front

If the discriminant is negative, what can we say about the roots?

Back

If the discriminant is negative, the quadratic equation has two complex (imaginary) roots.

Front

Solve the quadratic equation: 2x² + 3x - 5 = 0 using the quadratic formula.

Back

x = (−3 ± √(49)) / 4 = 1 and -2.5 (two distinct real roots).

Front

What is the value of the discriminant for the equation: 4x² + 4x + 1?

Back

D = 4² - 4(4)(1) = 16 - 16 = 0 (1 real root).

Front

How do you determine the number of real solutions from the discriminant?

Back

Count the number of real solutions based on the discriminant: D > 0 = 2 real solutions, D = 0 = 1 real solution, D < 0 = 0 real solutions.