2.5 Quadratic equations  (Page 5/14)

The discriminant

The quadratic formula    not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant    , or the expression under the radical, $b^2-4ac.$ The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. [link] relates the value of the discriminant to the solutions of a quadratic equation.

Using the pythagorean theorem

One of the most famous formulas in mathematics is the Pythagorean Theorem    . It is based on a right triangle, and states the relationship among the lengths of the sides as $a^2+b^2=c^2,$ where $a$ and $b$ refer to the legs of a right triangle adjacent to the $\mathrm{90°}$ angle, and $c$ refers to the hypotenuse. It has immeasurable uses in architecture, engineering, the sciences, geometry, trigonometry, and algebra, and in everyday applications.

We use the Pythagorean Theorem to solve for the length of one side of a triangle when we have the lengths of the other two. Because each of the terms is squared in the theorem, when we are solving for a side of a triangle, we have a quadratic equation. We can use the methods for solving quadratic equations that we learned in this section to solve for the missing side.

The Pythagorean Theorem is given as

where $a$ and $b$ refer to the legs of a right triangle adjacent to the ${90}^{\circ }$ angle, and $c$ refers to the hypotenuse, as shown in [link] .