Quadratic Equations and Functions

Quadratic Equation

A quadratic equation is a polynomial function of degree 2, which means it has at least one term with an exponent of 2. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

Where:

• $$a$$, $$b$$, and $$c$$ are real numbers.
• $$a$$ represents the leading coefficient, which is always positive.
• $$b$$ represents the coefficient of the linear term.
• $$c$$ represents the constant term.

Factors and Solutions

A quadratic equation can be written in the form of $$(x-p)(x-q) = (x-p)(x-q)$$, where $$p$$ and $$q$$ are the roots (or solutions). In the standard form, once the equation is written in the form $$ax^2 + bx + c = 0$$, the roots can be found using the quadratic formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

This formula gives two possible values for $$x$$, which are the roots of the quadratic equation.

Quadratic Functions

A quadratic function is a function that has a quadratic equation as its equation. It is a polynomial function of degree 2, which means it has a maximum or minimum point, depending on the sign of the leading coefficient. The graph of a quadratic function is a parabola, which is a symmetric shape that opens up if the leading coefficient is positive and down if the leading coefficient is negative.

Example

Consider the quadratic equation:

$$2x^2 + 3x - 4 = 0$$

Using the quadratic formula, the roots of the equation are:

$$x = \frac{-3 \pm \sqrt{(3)^2 - 4(2)(-4)}}{2(2)}$$

$$x = \frac{-3 \pm \sqrt{9 + 16}}{4}$$

$$x = \frac{-3 \pm \sqrt{25}}{4}$$

$$x = \frac{-3 \pm 5}{4}$$

So, the roots of the equation are:

$$x = \frac{-3 + 5}{4}$$

$$x = \frac{2}{4}$$

$$x = \frac{1}{2}$$

And:

$$x = \frac{-3 - 5}{4}$$

$$x = \frac{-8}{4}$$

$$x = -\frac{2}{2}$$

$$x = -1$$

These are the solutions of the quadratic equation.

Test your knowledge of quadratic equations, including their general form, roots, quadratic formula, and quadratic functions. Explore how to find the roots of a quadratic equation and understand the characteristics of quadratic functions such as parabolas.