JEE Main & Advanced Mathematics Complex Numbers and Quadratic Equations The Quadratic Expression

The Quadratic Expression

Category : JEE Main & Advanced

(1) Let \[f(x)\equiv a{{x}^{2}}+bx+c,\] \[a,b,c\in R\], \[a>0\] be a quadratic expression. Since, \[f(x)=a\,\left\{ {{\left( x+\frac{b}{2a} \right)}^{2}}-\left( \frac{{{b}^{2}}-4ac}{4{{a}^{2}}} \right) \right\}\] ……(i)

The following is true from equation (i)

(i) \[f(x)>0\,\,(<0)\] for all values of \[x\in R\] if and only if

\[a>0\,(<0)\] and \[D\equiv {{b}^{2}}-4ac<0\].

(ii) \[f(x)\ge 0\,(\le 0)\] if and only if

\[a>0\,(<0)\] and \[D\equiv {{b}^{2}}-4ac=0\].

In this case \[(D=0)\], \[f(x)=0\] if and only if \[x=-\frac{b}{2a}\]

(iii) If \[D\equiv {{b}^{2}}-4ac>0\] and \[a>0\,\,(<0),\] then

\[f(x)\,\left[ \begin{matrix} <0(>0), & \text{for }x\,\text{lying betw}\text{een the roots of }f(x)=0\,\,\,\,\,\,\,\, \\ >0(<0), & \text{for }x\,\text{not lying betw}\text{een the roots of }f(x)=0 \\ =0, & \text{for }x=\text{each of the roots of }f(x)=0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \\ \end{matrix} \right.\]

(iv) If \[a>0,\,(<0)\], then \[f(x)\] has a minimum (maximum) value at \[x=-\frac{b}{2a}\] and this value is given by

\[{{[f(x)]}_{\text{min}\,\text{(max)}}}=\frac{4ac-{{b}^{2}}}{4a}\].

(2) Sign of quadratic expression : Let \[f(x)=a{{x}^{2}}+bx+c\] or \[y=a{{x}^{2}}+bx+c\]

Where \[a,\,b,\,\,c\,\,\in R\] and \[a\ne 0,\] for some values of \[x,\,\,f(x)\] may be positive, negative or zero. This gives the following cases :

(i) \[a>0\] and \[D<0,\] so \[f(x)>0\] for all \[x\in R\] i.e., \[f(x)\] is positive for all real values of \[x\].

(ii) \[a<0\] and \[D<0,\] so \[f(x)<0\] for all \[x\in R\] i.e., \[f(x)\] is negative for all real values of \[x\].

(iii) \[a>0\] and \[D=0,\] so \[f(x)\ge 0\] for all \[x\in R\] i.e., \[f(x)\] is positive for all real values of \[x\] except at vertex, where \[f(x)=0\].

(iv) \[a<0\] and \[D=0,\] so \[f(x)\le 0\] for all \[x\in R\] i.e. \[f(x)\] is negative for all real values of \[x\] except at vertex, where \[f(x)=0\].

(v) \[a>0\] and \[D>0,\] let \[f(x)=0\] have two real roots \[\alpha \] and \[\beta \] \[(\alpha <\beta ),\] then \[f(x)>0\] for all \[x\in (-\infty ,\,\alpha )\cup (\beta ,\,\infty )\] and \[f(x)<0\] for all \[x\in (\alpha ,\,\beta )\].

(vi) \[a<0\] and \[D>0\], let \[f(x)=0\] have two real roots \[\alpha \] and \[\beta \]\[(\alpha <\beta )\]. Then x\[f(x)<0\] for all \[x\in (-\infty ,\,\alpha )\cup (\beta ,\,\infty )\] and \[f(x)>0\] for all \[x\in (\alpha ,\,\beta )\]

(3) Graph of a quadratic expression

We have \[y=a{{x}^{2}}+bx+c=f(x)\]

\[y=a\left[ {{\left( x+\frac{b}{2a} \right)}^{2}}-\frac{D}{4{{a}^{2}}} \right]\]\[\Rightarrow \] \[y+\frac{D}{4a}=a{{\left( x+\frac{b}{2a} \right)}^{2}}\]

Now, let \[y+\frac{D}{4a}=Y\] and \[X=x+\frac{b}{2a}\]

\[Y=a.{{X}^{2}}\]\[\Rightarrow \] \[{{X}^{2}}=\frac{1}{a}Y\]

(i) The graph of the curve \[y=f(x)\] is parabolic.

(ii) The axis of parabola is \[X=0\] or \[x+\frac{b}{2a}=0\]

i.e. (parallel to y-axis).

(iii) (a) If \[a>0,\] then the parabola opens upward.

(b) If \[a<0,\] then the parabola opens downward.