key term - Quadratic Equation

Definition

A quadratic equation is a polynomial equation of the second degree, where the highest exponent of the variable is 2. It takes the general form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are real numbers and $a$ is not equal to 0. Quadratic equations are fundamental in algebra and have important applications in various fields, including physics, engineering, and economics.

5 Must Know Facts For Your Next Test

1. Quadratic equations can be solved using various methods, including the square root property, completing the square, and the quadratic formula.
2. The square root property is used to solve quadratic equations of the form $x^2 = k$, where $k$ is a constant.
3. Completing the square is a method that involves transforming the quadratic equation into the form $(x - h)^2 = k$, where $h$ and $k$ are constants.
4. The quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, can be used to find the solutions of any quadratic equation.
5. Quadratic equations are also used in the distance and midpoint formulas, as well as in solving systems of nonlinear equations.

Review Questions

• Explain how the square root property can be used to solve a quadratic equation.
  • The square root property is used to solve quadratic equations of the form $x^2 = k$, where $k$ is a constant. To solve such an equation, we take the square root of both sides, resulting in $x = \pm \sqrt{k}$. This method is applicable when the quadratic equation can be easily rearranged into the form $x^2 = k$. It provides a straightforward way to find the solutions to the equation, as the square root of $k$ gives the two possible values of $x$.
• Describe the process of completing the square to solve a quadratic equation.
  • Completing the square is a method used to transform a quadratic equation into the form $(x - h)^2 = k$, where $h$ and $k$ are constants. This is done by adding and subtracting the square of half the coefficient of $x$, $\left(\frac{b}{2}\right)^2$, to the original equation. This allows the equation to be rewritten as a perfect square plus a constant, which can then be solved by taking the square root of both sides. Completing the square is a useful technique when the quadratic equation does not have a perfect square form or when the coefficients are not easily factored.
• Analyze the role of quadratic equations in solving systems of nonlinear equations.
  • Quadratic equations play a crucial role in solving systems of nonlinear equations, which involve equations of higher degrees, such as polynomials, exponentials, or trigonometric functions. When solving a system of nonlinear equations, one or more of the equations may be quadratic in nature. By isolating the quadratic equation and solving it using the methods discussed, such as the quadratic formula or completing the square, the solutions can be used to find the intersection points of the various curves represented by the system of equations. This allows for the determination of the points where the equations intersect, which is essential in solving systems of nonlinear equations.

Related terms

Parabola:

The graph of a quadratic equation is a parabola, a U-shaped curve that opens upward or downward depending on the sign of the coefficient $a$.

Discriminant:

The discriminant, denoted as $b^2 - 4ac$, is a value that determines the nature of the solutions to a quadratic equation, such as whether they are real, imaginary, or repeated.

Factoring:

Factoring is a method used to solve quadratic equations by expressing the equation as a product of two linear factors, which can then be set equal to zero to find the solutions.