A quadratic equation is a second-degree polynomial equation in a single variable x, with the highest exponent being 2, generally in the form ax² + bx + c = 0.
What are the components of a quadratic equation?
A quadratic equation has three main components: a (leading coefficient), b (linear coefficient), and c (constant term).
What is the Quadratic Formula?
The Quadratic Formula is x = (-b ± √(b² - 4ac)) / (2a), which is used to find the solutions of a quadratic equation.
What is the discriminant in a quadratic equation?
The discriminant is b² - 4ac and it determines the nature of the roots of the quadratic equation.
How does the discriminant indicate the nature of the roots?
If the discriminant is positive, there are two distinct real roots; if zero, there is one real root; if negative, there are two complex roots.
What does 'solving by factoring' mean for quadratic equations?
It involves expressing the quadratic equation as a product of its factors set to zero, then solving for x.
How can completing the square solve a quadratic equation?
This method involves creating a perfect square trinomial from the quadratic and solving for x.
When is the quadratic formula used in solving quadratic equations?
It is used when a quadratic equation cannot be easily factored.
What is an example of a quadratic equation?
An example is x² - 4x + 4 = 0.
How do you convert a standard quadratic equation to vertex form?
By completing the square and rearranging the equation into the form y = a(x - h)² + k.
Why are quadratic equations important in real-world applications?
They are used in physics, engineering, economics, and various sciences to model parabolic trajectories, optimize areas, and more.
What graph is represented by a quadratic equation?
A parabola, which may open upwards or downwards.
What determines the direction of the parabola in a quadratic equation graph?
The sign of the leading coefficient 'a'. If a > 0, the parabola opens upwards; if a < 0, it opens downwards.
What are the vertex and axis of symmetry in a quadratic function?
The vertex is the highest or lowest point of the parabola and the axis of symmetry is a vertical line through the vertex, x = -b/(2a).
How is the axis of symmetry related to the quadratic formula?
The axis of symmetry x = -b/(2a) is derived from the formula and represents the vertical line that divides the parabola into two symmetrical halves.