Quadratic Equations Summary

General Form of Quadratic Equations

• A quadratic equation occurs when a quadratic polynomial $p(x)$ is set to zero ($p(x) = 0$).
• The general form is $ax^2 + bx + c = 0$, where $a, b, c \in \mathbb{R}$ and $a \neq 0$.

Roots of a Quadratic Equation

• A root is the value of the variable that satisfies the equation such that $L.H.S = R.H.S$.
• Every quadratic equation has exactly two roots.
• If $x = 2$ and $x = 3$ are roots of $3x^2 - 2kx + 2m = 0$, their values can be substituted back into the equation to determine the constants $k$ and $m$.

Shreedharacharya's Rule and the Discriminant

• Shreedharacharya's Rule (the Quadratic Formula) provides the roots of $ax^2 + bx + c = 0$ using the formula: $x = \frac{-b \pm \sqrt{D}}{2a}$.
• The Discriminant ($D$) is defined as $D = b^2 - 4ac$.

Nature of Roots

• Real and distinct: Roots are real and different if $D > 0$ (e.g., $4x^2 - 5 = 0$ where $D = 80$).
• Real and equal: Roots are real and the same if $D = 0$ (e.g., $x^2 - 4x + 4 = 0$ where $D = 16 - 16 = 0$).
• No real roots: No real solutions exist if $D < 0$ (e.g., $x^2 + x + 1 = 0$ where $D = -3$).

Solving by Factorization

• Equations can be solved by grouping and factoring, such as splitting the middle term in $8x^2 - 22x - 21 = 0$ into $-28x + 6x$.
• Simple quadratic forms like $3x^2 - 2 = 0$ can be solved by isolating the variable to find $x = \pm \sqrt{\frac{2}{3}}$.
• Radical equations like $\sqrt{3}x^2 + 10x + 7\sqrt{3} = 0$ and expressions involving powers like $(x - 1)^2 - 5(x - 1) - 6 = 0$ are also solvable through factorization techniques.