Quadratic Equations: Factoring, Formula, Discriminant, Vertex Form, and Graphing

10 Questions

Как называется процесс разложения квадратного уравнения на более простые выражения?

Факторизация

Какое значение вычисляется с помощью дискриминанта в квадратном уравнении?

$b^2 - 4ac$

Как называется общее решение квадратного уравнения $ax^2 + bx + c = 0$?

Квадратная формула

Если дискриминант квадратного уравнения равен нулю, сколько действительных корней у этого уравнения?

Один корень

Чему равно раскрытие выражения $(x + 3)^2$?

$x^2 + 6x + 9$

Что такое форма вершины квадратного уравнения?

Способ записи квадратного уравнения для явного отображения вершины параболы

Как изменяется форма параболы, если коэффициент 'a' в квадратном уравнении положительный?

Открывается вверх

Где находятся точки пересечения параболы с осью 'x'?

Точки пересечения графика с осью абсцисс

Как можно найти координату 'x' вершины параболы, заданной в форме '(x - h)^2 + k'?

Поставив 'x = h'

Что представляют собой точки пересечения параболы с осями координат?

Точки, где график пересекает оси координат

Study Notes

Quadratic Equations

Quadratic equations are mathematical expressions that involve the square of a variable (x) and at least one other term. They are typically written in the form ax^2 + bx + c = 0, where a, b, and c are constants. These equations have a rich history and are used in various mathematical contexts. In this article, we will discuss the subtopics of factoring, the quadratic formula, the discriminant, vertex form, and graphing quadratic equations.

Factoring

Factoring is the process of breaking down a quadratic equation into simpler expressions. It involves finding the factors of the quadratic equation and expressing it as a product of two binomials. For example, the equation x^2 + 4x + 4 = 0 can be factored to (x + 2)^2 = 0, which reveals that x = -2 is a solution.

Quadratic Formula

The quadratic formula is a general solution to a quadratic equation ax^2 + bx + c = 0. It is given by:

x = (-b ± √(b^2 - 4ac)) / 2a

For example, the equation 2x^2 + 5x + 3 = 0 has the solutions (1 ± √13) / 4.

Discriminant

The discriminant is a value obtained from the quadratic formula and helps determine the nature of the roots of a quadratic equation. It is calculated as b^2 - 4ac. If the discriminant is greater than 0, the equation has two distinct real roots. If the discriminant is equal to 0, the equation has one real root. If the discriminant is less than 0, the equation has two complex roots.

Vertex Form

Vertex form is a way of writing a quadratic equation to explicitly show the vertex (the highest or lowest point) of the parabola it represents. The vertex form of a quadratic equation ax^2 + bx + c = 0 is given by a(x - h)^2 + k, where h and k are constants. This form is particularly useful for graphing quadratic functions, as it allows us to easily find the x-coordinate of the vertex.

Graphing

Graphing quadratic equations involves plotting the points on a coordinate plane and connecting them with a curve. The shape of the curve depends on the sign of the coefficient a:

When a > 0, the curve opens upwards.
When a < 0, the curve opens downwards.

The graph of a quadratic equation also contains the vertex, x-intercepts, and y-intercept. The x-intercepts are the points where the curve crosses the x-axis, and the y-intercept is the point where the curve crosses the y-axis. The vertex is the highest or lowest point of the curve, and its x-coordinate can be found by setting x = h in the vertex form of the equation.

In conclusion, quadratic equations play a crucial role in mathematics, and understanding their properties and behaviors is essential for solving a wide range of problems. Factoring, the quadratic formula, discriminant, vertex form, and graphing are all important techniques and concepts that help us work with and visualize these equations.