Problem 155 - College Algebra

Question

What is a quadratic equation?

Step-by-Step Solution

Answer

A quadratic equation is a second-order polynomial equation in the standard form \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are constants. Its graph forms a parabola, and its solutions can be applied in a range of real-life problems.

Step 1: Definition

A quadratic equation is a second-order polynomial equation that can be expressed in the standard form \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are constants and \(a\) does not equal zero.

Step 2: Details and Real-world Applications

Values of \(x\), for which the equation equals zero, are known as solutions or roots of the equation. Quadratic equations and their solutions appear in diverse real-life situations such as calculation of areas, determination of an object's maximum height in physics, and optimization in operations research.

Step 3: Relationship with Parabolas

Graphically, a quadratic equation describes a parabola, the shape of which depends on the coefficients \(a\), \(b\), and \(c\). When \(a > 0\), the parabola opens upwards like a 'U'. When \(a < 0\), the parabola opens downwards like an 'n'.