Algebra: Quadratic Equations

What is a quadratic equation?

An equation of degree 2, meaning the highest power of the variable is 2.

What is the standard form of a quadratic equation?

ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.

How many roots (solutions) does a quadratic equation have?

A quadratic equation always has exactly two roots, which can be real or complex, and may sometimes be equal.

Name three common methods for solving quadratic equations.

Factoring, using the Quadratic Formula, and Completing the Square.

What is the Quadratic Formula?

x = [-b ± sqrt(b² - 4ac)] / 2a.

When is the Quadratic Formula particularly useful for solving quadratic equations?

It is useful when factoring is difficult or impossible, as it can solve any quadratic equation.

What is the discriminant of a quadratic equation and what is its formula?

The discriminant is the expression b² - 4ac, found under the square root in the Quadratic Formula.

What does a positive discriminant (b² - 4ac > 0) tell you about the roots of a quadratic equation?

There are two distinct real roots.

What does a discriminant equal to zero (b² - 4ac = 0) tell you about the roots of a quadratic equation?

There is exactly one real root (a repeated root).

What does a negative discriminant (b² - 4ac < 0) tell you about the roots of a quadratic equation?

There are two distinct complex (non-real) conjugate roots.

What is the graph of a quadratic equation called?

A parabola.

What is the vertex of a parabola?

The turning point of the parabola, which is either its minimum or maximum point.

How do you find the x-coordinate of the vertex of a parabola given in standard form (ax² + bx + c)?

The x-coordinate of the vertex is given by the formula x = -b / 2a.

What are the roots of a quadratic equation also known as on its graph?

They are also known as the x-intercepts or zeros of the function.

If r1 and r2 are the roots of ax² + bx + c = 0, what is their sum (r1 + r2) in terms of a, b, c?

The sum of the roots is -b/a.