Quadratic Equations

Quadratic Equations

A quadratic equation is a second-degree polynomial equation of the form

ax^2 + bx + c = 0

. Its graph is a symmetric U-shaped curve known as a parabola.

The Standard Form

ax^2 + bx + c = 0 \quad (a \neq 0)

Factoring: Best when the trinomial is easily reducible to $(x-p)(x-q)$ .
Completing the Square: Useful for deriving the vertex and for non-factorable equations.
Quadratic Formula: The universal "hammer" that works for all quadratics.

Universal formula for finding roots of any quadratic equation.

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Variables

Symbol	Description	Unit
$x$	The roots or solutions of the quadratic equation	-
$a$	Coefficient of the squared term (cannot be zero)	-
$b$	Coefficient of the linear term	-
$c$	Constant term	-
$b^2 - 4ac$	The discriminant (determines the nature of the roots)	-

Vieta's Formulas

Vieta's Formulas relate the roots of a polynomial to its coefficients. For a quadratic equation

ax^2 + bx + c = 0

with roots

r_1

and

r_2

Sum of the Roots ( $r_1 + r_2$ ): equals $-b/a$ .
Product of the Roots ( $r_1 \cdot r_2$ ): equals $c/a$ .
Difference of the Roots ( $|r_1 - r_2|$ ): equals $\frac{\sqrt{b^2 - 4ac}}{|a|}$ . This can be derived from the identity $(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1r_2$ .
Sum of Squares of Roots ( $r_1^2 + r_2^2$ ): equals $\frac{b^2 - 2ac}{a^2}$ . This is derived from $(r_1 + r_2)^2 - 2r_1r_2$ .
Constructing an Equation: If you already know the roots, you can quickly build the quadratic equation: $x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0$ .

\Delta

Determines the number and nature of roots based on the expression

b^2 - 4ac

Interactive Visualizer

Adjust

a

to control width and direction,

b

to shift the axis of symmetry, and

c

to change the y-intercept.

a (Quadratic)1

b (Linear)0

c (Constant)0

y = x^2

Vertex: (0.00, 0.00)

Key Takeaways

Vertex ( $h, k$ ): The peek or valley of the parabola. $h = -b/(2a)$ .
Symmetry: Every parabola is symmetric about the vertical line $x = h$ .
Complex Roots: If the discriminant is negative, the graph never touches the x-axis.
Applications: Projectile motion, area optimization, and profit maximization often involve quadratics.

The Standard Form