Quadratic Formula & Discriminant Review

11 Q&A flashcards covering the quadratic formula, the discriminant, graph interpretation, identifying coefficients, and parameter‐based root conditions.

What is the quadratic formula for solving ax² + bx + c = 0?

x = (-b ± √(b² - 4ac)) / (2a).

List the steps for solving a quadratic equation using the quadratic formula.

1) Write the equation in standard form ax² + bx + c = 0 and identify a, b, c. 2) Substitute these values into x = (-b ± √(b² - 4ac)) / (2a). 3) Evaluate the discriminant b² - 4ac. 4) Simplify the expression to obtain the roots, leaving answers in surd form if appropriate.

What part of the quadratic formula is called the discriminant?

The expression under the square root sign, b² - 4ac.

How does the discriminant determine the number of real roots of a quadratic equation?

b² - 4ac > 0 → two distinct real roots; b² - 4ac = 0 → one repeated real root; b² - 4ac < 0 → no real roots.

How does the sign of the discriminant affect the graph of a quadratic function?

Positive discriminant: parabola crosses the x-axis at two points. Zero discriminant: parabola touches the x-axis at one point (vertex). Negative discriminant: parabola does not intersect the x-axis.

In the quadratic expression ax² + bx + c, what are a, b, and c?

a is the coefficient of x², b is the coefficient of x, and c is the constant term.

What inequality must be satisfied for a quadratic to have two real roots?

b² - 4ac > 0.

What condition gives exactly one real root for a quadratic?

b² - 4ac = 0.

What condition gives no real roots for a quadratic?

b² - 4ac < 0.

Describe the method for finding the range of a parameter (e.g., k) that ensures a quadratic has a specified number of real roots.

Rewrite the discriminant b² - 4ac in terms of the parameter, set the appropriate inequality or equality (>, =, < 0) based on the desired number of roots, and solve that inequality for the parameter.

Example: For x² + 4kx + 4 = 0 to have two real roots, what condition must k satisfy?

b = 4k, a = 1, c = 4 → Discriminant: (4k)² - 4·1·4 = 16k² - 16 > 0 → k² > 1 → k < –1 or k > 1.