Quadratic Equations

A Quadratic Equation is an algebraic equation of the second degree, having the standard form $ax^2 + bx + c = 0$, where $x$ is the variable and $a, b, c$ are real numbers with $a \neq 0$. In board exams, you must be able to recognize this standard form and reduce equations that initially appear complex—such as fractional or radical equations—into this standard format. Always remember that the term $a \neq 0$ is a crucial condition; if $a = 0$, the equation degenerates into a linear equation.

There are three primary methods taught in the curriculum to solve quadratic equations, and board papers frequently test all of them: Factorization, Completing the Square, and using the Quadratic Formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. While factorization is the quickest method when roots are rational, the quadratic formula is universally applicable and a safe choice when numbers are complex or irrational. Students are often explicitly instructed in exam questions to use a specific method, so practice with all three is essential.

A vital recurring topic in BISE examinations is the Discriminant, given by the expression $\Delta = b^2 - 4ac$. Examiners frequently test your ability to determine the nature of the roots (real/imaginary, rational/irrational, and equal/unequal) without actually solving the equation. Furthermore, questions heavily feature Vieta's Formulas—the relationship between the roots and coefficients of a quadratic equation—allowing you to find sum and product of roots, construct new quadratic equations given specific roots, and solve simultaneous equations or radical equations reducible to quadratic form.