Quadratics are polynomial equations of the second degree, meaning they include at least one squared term. These equations are also referred to as quadratic equations. The general representation of a quadratic equation is:
ax² + bx + c = 0
Here, x is the unknown variable, and a, b, and c are numerical coefficients. For instance, x² + 2x + 1 is an example of a quadratic equation. Note that a ≠ 0 because if a = 0, the equation reduces to a linear equation, such as:
bx + c = 0
This would no longer be considered a quadratic equation.
The coefficients a, b, and c are often referred to as the quadratic coefficients.
Roots or Zeros of Quadratic Equations
The solutions to a quadratic equation are the values of x that satisfy the equation. These solutions are known as the roots or zeros of the quadratic equation. For any polynomial, the roots are the solutions that make the equation true.
What is a Quadratic Equation?
A quadratic equation, also called a quadratic, is a polynomial equation where the highest degree is two. It is typically written as:
ax² + bx + c = 0
Here, x is the variable, and a, b, and c are constants.
Standard Form of a Quadratic Equation
Since quadratic equations involve only one variable, they are univariate equations. The variable x has a non-negative integer power, and the highest power is 2. Therefore, it is a second-degree polynomial.
The solutions, or zeros, of the quadratic equation satisfy the equation when substituted for x. Quadratic equations have two roots, and substituting these roots into the equation will make the left-hand side equal to zero.
Quadratic Formula
To find the roots of a quadratic equation, the quadratic formula is used. Given a quadratic equation ax² + bx + c = 0, the roots can be calculated using the formula:
x = (-b ± sqrt(b² - 4ac))/2a
The ± sign indicates that there are two possible solutions for x.
Examples of Quadratic Equations
Below are some examples of quadratic equations in the standard form ax² + bx + c = 0:
- x² – x – 9 = 0
- 5x² – 2x – 6 = 0
- 3x² + 4x + 8 = 0
- –x² + 6x + 12 = 0
Examples of quadratic equations without the constant term c:
- –x² – 9x = 0
- x² + 2x = 0
Methods to Solve Quadratic Equations
There are four primary methods to solve quadratic equations:
1. Factoring
2. Completing the Square
3. Using the Quadratic Formula
4. Taking the Square Root
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