Quadratic Equations Study Guide
1. Basics of Quadratic Equations
● Definition: A quadratic equation is a second-degree polynomial equation in a
single variable, with the general form
●
ax2+bx+c=0
● ax
● 2
● +bx+c=0.
● Standard Form:
●
ax2+bx+c=0
● ax
● 2
● +bx+c=0
● Discriminant (
Δ
● Δ):
● Δ=b2−4ac
●
● Δ=b
● 2
● −4ac
2. Solving Quadratic Equations
● Factoring: Factor the quadratic expression and set each factor equal to zero.
● Quadratic Formula:
x=−b±Δ2a
● x=
●
● 2a
● −b±
● Δ
●
●
●
● Completing the Square: Convert the equation to the form
(x−h)2=k
● (x−h)
●
● 2
● =k and solve for
●
x
● x.
3. Graphical Representation
● Graphing Quadratic Equations: Understand the shape and key features of the
parabolic graph.
● Vertex Form:
y=a(x−h)2+k
● y=a(x−h)
●
● 2
● +k
● Vertex: The point
●
(h,k)
● (h,k) is the vertex of the parabola.
4. Applications of Quadratic Equations
● Word Problems: Practice solving real-life problems using quadratic equations.
● Maximum and Minimum Values: Understand how to find the maximum or
minimum value of a quadratic function.
5. Special Cases
● Perfect Square Trinomials:
●
a2+2ab+b2=(a+b)2
● a
● 2
● +2ab+b
● 2
●
=(a+b)
● 2
●
● Difference of Squares:
a2−b2=(a+b)(a−b)
● a
●
● 2
● −b
● 2
● =(a+b)(a−b)
● Completing the Square: Transforming a quadratic expression to a perfect square
trinomial.
6. Quadratic Inequalities
● Solving Quadratic Inequalities: Determine the intervals for which the inequality is
satisfied.
● Graphical Representation of Inequalities: Understand how to represent quadratic
inequalities on a number line.