Quadratic Equation

Quadratic Equations — JEE Mathematics Complete Guide

Standard form, discriminant and roots, Vieta, vertex and range, inequalities, parameter questions, classic patterns, pitfalls — plus an interactive graph. Fully blogger‑friendly with proper equation rendering.

JEE syllabus coverage

• Standard form: $ax^2+bx+c=0,\; a\ne 0$
• Discriminant & roots: $D=b^2-4ac$; nature of roots; quadratic formula
• Vieta’s relations: sum/product of roots; forming equations
• Vertex, axis, range: completing square; min/max; graph
• Inequalities: sign chart using roots and $a$
• Parameter questions: conditions on $k$ for roots/restrictions

Core results

Quadratic formula and nature

• $x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
• $D>0$: two distinct real; $D=0$: equal real; $D<0$: non‑real conjugates

Vieta’s relations

• If roots are $\alpha,\beta$: $\alpha+\beta=-\dfrac{b}{a}$, $\alpha\beta=\dfrac{c}{a}$
• Equation from roots (leading coefficient $a$): $a\left(x^2-(\alpha+\beta)x+\alpha\beta\right)=0$

Vertex and range

• $ax^2+bx+c=a\left(x+\dfrac{b}{2a}\right)^2-\dfrac{D}{4a}$
• Vertex: $\left(-\dfrac{b}{2a},-\dfrac{D}{4a}\right)$; Range: $a>0\Rightarrow [y_v,\infty)$, $a<0\Rightarrow (-\infty,y_v]$

Inequalities (exam‑style)

• $a>0$ opens up (U); $a<0$ opens down (∩).
• $D>0$, roots $x_10$: $f(x)\ge 0$ on $(-\infty,x_1]\cup[x_2,\infty)$, and $f(x)\le 0$ on $[x_1,x_2]$. Reverse for $a<0$.
• $D=0$: touches x‑axis at $x_v$. If $a>0$: $f(x)\ge 0$ for all $x$; if $a<0$: $f(x)\le 0$ for all $x$.
• $D<0$: no real roots. If $a>0$: $f(x)>0$; if $a<0$: $f(x)<0$.

For rational inequalities, exclude where denominator $=0$ and apply sign chart by intervals.

Parameter (k) frames

Reality/equality of roots

• Given $ax^2+bx+c(k)=0$: define $D(k)=b^2-4a\,c(k)$. Real roots $\Leftrightarrow D(k)\ge 0$; equal roots $\Leftrightarrow D(k)=0$.

Both roots on one side of $m$

• Let $g(x)=f(x+m)=ax^2+Bx+C$, $B=b+2am,\; C=am^2+bm+c$.
• For $a>0$: both $>m \Leftrightarrow -\dfrac{B}{a}>0$ and $\dfrac{C}{a}>0$; both $0$.

Interactive quadratic graph

a: 1.0 b: 0.0 c: 0.0 Shade: f(x) ≥ 0 f(x) ≤ 0

D = … Roots: … Vertex: …

Axes have arrowheads; vertex (•), roots (if real), axis of symmetry (dashed), and shaded inequality regions.

Formula crib sheet

Result	Formula	Use
Quadratic formula	$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$	Exact roots
Discriminant	$D=b^2-4ac$	Nature of roots
Vieta	$\alpha+\beta=-\dfrac{b}{a},\;\alpha\beta=\dfrac{c}{a}$	Forming equations
Vertex	$\left(-\dfrac{b}{2a},-\dfrac{D}{4a}\right)$	Range, extrema
Completed square	$a\left(x+\dfrac{b}{2a}\right)^2-\dfrac{D}{4a}$	Graph/transform

Practice set (quick keys)

1. Minimum of $2x^2-8x+7$.
   Key: $x_v=2$, $y_{\min}=-\dfrac{D}{4a}=-1$.
2. Find $k$ so $x^2-4x+k=0$ has equal roots.
   Key: $D=16-4k=0\Rightarrow k=4$.
3. Solve $x^2-5x+6\ge 0$.
   Key: roots $2,3$; solution $(-\infty,2]\cup[3,\infty)$.