ax² + bx + c
A quadratic equation, also known as a quadratic equation, is a polynomial equation of degree two that can be expressed in standard form:
Note that the coefficient a cannot be zero, because if it were, the polynomial equation would no longer be of the second degree and would become a first-degree (linear) equation.
The roots of a quadratic equation are the values of x that satisfy the equation ax² + bx + c = 0.
Quick example: In the equation x² - 5x + 6 = 0, the roots are x₁ = 2 and x₂ = 3, as substituting these values into the equation gives us 0.
Solving a quadratic equation involves finding the values of x that satisfy the equation in the standard form ax² + bx + c = 0. Here is a step-by-step guide to solve this type of equation:
By following these steps, you will be able to solve any quadratic equation and find its roots.
To use the Quadratic Equation Solver, follow the steps below:
After clicking the button, the calculator will display results for the entered equation, including the equation roots, the graph, and other relevant information. Additionally, graphs will be presented that illustrate the shape of the parabola represented by the quadratic equation.
Suppose you have the quadratic equation x² - 5x + 6 = 0.
By entering these values in the solver and clicking "Calculate Roots", you will obtain the roots x₁ = 2 and x₂ = 3, and the delta (Δ) = 1. Additionally, the graph of the parabola corresponding to the equation will also be displayed.
The graph of a quadratic equation is a parabola, which can open upward or downward depending on the value of coefficient 'a'. If 'a' is positive, the parabola opens upward; if 'a' is negative, the parabola opens downward.
The parabola is symmetric with respect to a vertical line called the axis of symmetry, which passes through the vertex of the parabola. This means that points equidistant from the axis of symmetry have the same y-value.
Additionally, it is possible to identify important points on the graph, such as the vertex, which is the highest or lowest point of the parabola. The vertex can be calculated using the formulas:
Factoring a quadratic equation is a method used to express the equation in the form of a product of two binomials. The factoring is of the form:
Quick example: The equation x² - 5x + 6 = 0 factors as (x - 2)(x - 3) = 0. Note that the product (x - 2)(x - 3) equals 0 both if (x - 2) = 0 and if (x - 3) = 0, which makes it easy to see that the roots are x₁ = 2 and x₂ = 3.
Another quick example: In addition to being easy to find roots in factored form, it is also possible to find the factored form from the roots. For example, if we have the roots x₁ = 4 and x₂ = -1, we can set up the factored form as (x - 4)(x + 1) = 0. For curiosity, expanding this product, we get the equation x² - 3x - 4 = 0.
Factoring is a useful tool for solving quadratic equations, especially when the roots are rational numbers. It is also interesting to note that if the roots are complex, the factoring can be expressed in terms of complex numbers. Additionally, if the delta (Δ) is 0, the factoring will result in a squared binomial.
Quadratic equations are widely used in various areas of real life, including physics, engineering, economics, and biology. They are particularly useful for modeling situations involving motion, growth, and optimization.
Some practical examples include: trajectories of objects in free fall, calculation of maximum areas, analysis of profits and losses, and modeling of biological populations. Quadratic equations help describe and predict complex behaviors in natural and artificial systems.