In algebra, an equation is a statement that shows that two expressions are equal. Equations are used to solve problems and find unknown variables.
A first order equation is an equation that contains only one variable that is raised to the first power (also known as linear equations). For example, the equation 2x + 3 = 7 is a first order equation because the variable x is raised to the first power.
A second order equation, on the other hand, contains one variable that is raised to the second power (also known as quadratic equations). For example, the equation x^2 + 3x + 2 = 0 is a second order equation because the variable x is raised to the second power.
The solutions to first order equations can be found by isolating the variable on one side of the equation. For example, to solve the equation 2x + 3 = 7, we can subtract 3 from both sides of the equation to get:
2x = 4
Then, we can divide both sides of the equation by 2 to get:
x = 2
The solutions to second order equations are found using the quadratic formula, which is a formula that gives the solutions to any second order equation. The quadratic formula is:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a, b, and c are constants in the equation ax^2 + bx + c = 0.
For example, to solve the equation x^2 + 3x + 2 = 0 using the quadratic formula, we can identify that a = 1, b = 3, and c = 2. Substituting these values into the quadratic formula, we get:
x = (-3 ± sqrt(3^2 - 4(1)(2))) / 2(1)
This simplifies to:
x = (-3 ± sqrt(1)) / 2
x = (-3 ± 1) / 2
So the solutions to the equation are:
x = -2 or x = -1