Quadratic Formula and Discriminant

Front

What is the Quadratic Formula?

Back

The Quadratic Formula is used to find the solutions 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?

Back

The Discriminant (D = b² - 4ac) indicates the nature of the roots of a quadratic equation. If D > 0, there are two distinct real solutions; if D = 0, there is one real solution; if D < 0, there are no real solutions.

Front

How do you determine if a parabola opens upwards or downwards?

Back

A parabola opens upwards if the coefficient of x² (a) is positive, and it opens downwards if a is negative.

Front

Identify a, b, and c in the equation 4x² - 8x = 3.

Back

a = 4, b = -8, c = -3 (after rearranging to standard form: 4x² - 8x - 3 = 0).

Front

What is the significance of a negative Discriminant?

Back

A negative Discriminant indicates that the quadratic equation has no real solutions, meaning the graph does not intersect the x-axis.

Front

If the Discriminant is zero, how many solutions does the quadratic equation have?

Back

If the Discriminant is zero, the quadratic equation has exactly one real solution (a repeated root).

Front

What is the vertex form of a quadratic equation?

Back

The vertex form of a quadratic equation is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Front

How can you find the vertex of a parabola given in standard form?

Back

The vertex (h, k) can be found using h = -b/(2a) and k = f(h), where f(x) is the quadratic function.

Front

What does it mean if a quadratic equation has two real solutions?

Back

It means the graph of the quadratic intersects the x-axis at two distinct points.

Front

What is the standard form of a quadratic equation?

Back

The standard form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants.