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.