Chapter 4: Quadratic Equations

A quadratic equation in $x$ is one of the form $a x^2 + b x + c = 0$ with $a \ne 0$. After polynomials (Chapter 2), this is the next big algebra topic. The difference is that we no longer just describe zeroes , we now have machinery to find them, reliably, every time.

Three techniques run the chapter. Factorisation is fastest when the quadratic splits neatly. Completing the square turns any quadratic into a "perfect square plus constant" form. The quadratic formula , born from completing the square , solves every quadratic uniformly. Alongside, the discriminant $D = b^2 - 4 a c$ tells you, without solving, how many real roots the equation has.

For boards, this chapter is dense. Expect $1$-mark MCQs on discriminant and nature of roots, $2$-mark factorisation, $3$-mark solving by the formula, and $4$- to $5$-mark word problems on speed, time, geometry, and consecutive-number puzzles. Quadratics also drive projectile motion in physics, profit curves in economics, and parabolic shapes in engineering.

By the end you should be able to look at any quadratic and decide instantly: factor it, complete the square, or punch into the formula. The choice is a matter of speed, not correctness.

What's inside

• Standard form and identification , what counts as a quadratic.
• Solution by factorisation , splitting the middle term.
• Completing the square , a derivation that flows into the quadratic formula.
• Quadratic formula and the discriminant , the universal solver.
• Nature of roots , real, equal, distinct, or complex.
• Word problems , applying quadratics to real situations.

Key results / Formula card

For $a x^2 + b x + c = 0$, $a \ne 0$: