Solution of Linear and Quadratic Equations
Solution of Linear and Quadratic Equations
Linear Equations
- A linear equation is an equation where the highest degree (or power) of the variable (commonly
x
) is one. - Linear equations have the form
ax + b = 0
; wherea
is the coefficient ofx
,b
is the constant anda ≠ 0
. - Solutions of linear equations are found by isolating
x
on one side of the equation.
Quadratic Equations
- A quadratic equation is an equation in which the highest degree (or power) of the variable (commonly
x
) is2
. - Quadratic equations have the form
ax^2 + bx + c = 0
; wherea
is the coefficient ofx^2
,b
is the coefficient ofx
,c
is the constant anda ≠ 0
. - There are three methods for solving quadratic equations: factoring, using the quadratic formula, and completing the square.
Factoring
- Factoring to solve for a quadratic equation is the process of writing the equation as two binomial expressions set equal to zero. Once factored, you can set each factor equal to zero and solve for
x
. - Factoring is best used when all terms are divisible by a common factor or when the equation can be written as a perfect square trinomial.
Quadratic Formula Method
- The Quadratic Formula:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
. This method is beneficial for all types of quadratic equations, especially when factoring is not favorable.
Completing the Square
- Completing the square method involves rewriting the quadratic equation so that the left-hand side is a perfect square trinomial. This will allow you to write the equation in the form
(x - p)^2 = q
. - This method allows us to derive the quadratic formula and is useful for understanding the graphical transformation of quadratic functions.
Remember: All three methods should lead to the same solutions for x
. It is critical to familiarize yourself with each method so that you can select the most efficient method based on the given quadratic equation.