Ch 04 - Quadratic Equations10th Mathematics

Standard Form

A quadratic equation in the variable $x$ is strictly written in the descending order of its degrees as $ax^2 + bx + c = 0$.

Boundary Conditions / Domain

• The leading coefficient of $x^2$ must strictly be non-zero ($a \neq 0$).
• If $a = 0$, the $x^2$ term vanishes, mathematically reducing it to a linear equation.
• Coefficients $a, b$, and $c$ must be real numbers.

Complete Quadratic Equation

Takes the form $ax^2 + bx + c = 0$ where all coefficients are strictly non-zero ($a \neq 0, b \neq 0, c \neq 0$).

Pure Quadratic Equation

Takes the specific form $ax^2 + c = 0$.

• It strictly lacks the linear $x$ term because its coefficient is zero ($b = 0$).
• Significance: Pure quadratic equations don't require the quadratic formula or splitting the middle term; they can be solved by direct square rooting.

Zeroes

Associated with a quadratic polynomial $p(x) = ax^2 + bx + c$.
It is the mathematical value of $x$ that makes the polynomial evaluate to zero.

Roots (or Solutions)

Associated with a quadratic equation $ax^2 + bx + c = 0$.
It is the value of $\alpha$ such that $a\alpha^2 + b\alpha + c = 0$.

The Connection

The zeroes of a quadratic polynomial are mathematically identical to the roots of its corresponding quadratic equation.