Quadratic Equations: Vertex, Discriminant, Factoring, Formula & Graphing

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

Quadratic equations are mathematical expressions involving a second degree polynomial function of one variable. They have the general form ax^2 + bx + c = 0, where a is the coefficient of the squared term, b is the coefficient of the linear term, and c is the constant term. These coefficients serve as important information when analyzing quadratic equations. Let's explore some key aspects of quadratic equations: vertex form, discriminant, factoring, quadratic formula, and graphing.

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Vertex Form

The vertex form of a quadratic equation is often used for parabolas that open vertically, meaning they are symmetric around the y-axis. The equation in vertex form is given by (x - a)^2 = b, where (x) is the independent variable. In this form, the vertex of the parabola corresponds to ((a, b)), and the axis of symmetry passes through it. This can be useful when graphing quadratic functions, as it allows us to find the exact location of the vertex and the point at which the function attains its minimum or maximum value.

Example: Consider the equation (f(x) = x^2 + 2x + 1). To rewrite this equation in vertex form, we factor out the perfect square trinomial: (f(x) = (x+1)^2). Now, comparing the standard form of the equation with the vertex form, we see that (a = -1) and (b = 1). Therefore, the vertex form of the equation (f(x) = x^2 + 2x + 1) is (x^2 + 2x + 1 = (-1)^2).

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Discriminant

In quadratic equations, the discriminant is a quantity derived from the coefficients of a quadratic equation to determine the number and nature of the roots of the equation. The discriminant is calculated using the formula b^2 - 4ac. If this value is positive, there are two distinct real roots; if it is zero, there is one repeated root (also known as a double root); and if it is negative, there are complex conjugate roots. These different scenarios can be analyzed through various methods depending on the nature of the quadratic equation.

Example: For the equation (x^2 + 5x + 6 = 0), we find that the discriminant is ((5)^2 - 4(1)(6) = 25 - 24 = 1). Since the discriminant is positive, there are two distinct real roots. To solve for these roots, we can use the quadratic formula or factor the equation.

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Factoring

Factoring quadratic equations involves rewriting them by breaking down their terms into factors that multiply together to form the original expression. This can sometimes simplify solving quadratic equations or provide useful insights about the behavior of the function. Some common factoring techniques include factoring out perfect squares, recognizing trinomials of the form (a^2 + 2ab + b^2), and finding common multiples between the first and last terms of the equation.

Example: Consider the equation (x^2 + 6x + 9 = 0). We notice that this can be factored as ((x+3)^2 = 0). This means that (x=-3) is one solution to the equation.

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Quadratic Formula

The quadratic formula allows us to find the solutions to any quadratic equation, regardless of its form or coefficients. It states that if we have an equation of the form ax^2 + bx + c = 0, then the solutions are given by:

[ x=\frac{-b±\sqrt{b^2-4ac}}{2a} ]

Here, (a), (b), and (c) are the coefficients of the quadratic equation. The first term under the square root, (b^2), is the square of the coefficient of the linear term in the equation. The second term, (4ac), is the product of the coefficient of the squared term and the constant term. The second term under the square root, (b^2 - 4ac), is known as the discriminant.

Example: For the equation (x^2 + 5x + 6 = 0), we find that the discriminant is ((5)^2 - 4(1)(6) = 25 - 24 = 1). Using the quadratic formula, we obtain the solutions:

[ x=\frac{-5 ± \sqrt{1}}{2(1)} = \frac{-5 ± 1}{2} ]

This gives us the solutions (x=4) and (x=-3), which we can also find by factoring the equation.

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Graphing

Graphing quadratic equations involves plotting the function's points and visualizing the behavior of the function over a given interval. The shape of the graph depends on the values of the coefficients in the equation. For example, if the leading coefficient (the coefficient of the squared term) is positive, the graph is a parabola that opens upward, and if it is negative, the graph is a parabola that opens downward. The vertex of the parabola is the point where the graph achieves its minimum or maximum value.

Example: Consider the equation (f(x) = x^2 + 2x + 1). To graph this function, we first find the vertex:

[ x = \frac{-2}{2} = -1 ] [ y = f(-1) = (-1)^2 + 2(-1) + 1 = 1 ]

Thus, the vertex is ((-1, 1)). To plot the points on the graph, we can use the quadratic formula or factor the equation. For example, we can find the x-intercepts by setting (y=0) and solving for (x) using the quadratic formula:

[ x=\frac{-2 ± \sqrt{2^2-4(1)(1)}}{2(1)} = \frac{-2 ± \sqrt{0}}{2} = \frac{-2}{2} = -1 ]

This gives us the x-intercepts ((-1, 0)) and ((1, 0))[6