QUADRATIC EQUATION
1. SOLUTION OF QUADRATIC EQUATION & RELATION BETWEEN ROOTS & CO-EFFICIENTS :
(a) The solutions of the quadratic equation, \(a x^{2}+b x+c=0\) is given by \(\mathbf{x =\dfrac{- b \pm \sqrt{ b ^{2}-4 a c }}{ 2 a }}\)
(b) The expression \(b^{2}-4 a c \equiv D\) is called the discriminant of the quadratic equation.
(c) If \(\alpha\, \& \, \beta\) are the roots of the quadratic equation \(a x^{2}+b x+c=0\), then;
(i) \(\alpha+\beta=-b / a\) (ii) \(\alpha \beta= c / a\) (iii) \(|\alpha-\beta|=\sqrt{ D } /| a |\)
(d) Quadratic equation whose roots are \(\alpha \) & \(\beta\) is \((x-\alpha)(x-\beta)=0\)
i.e. \(x ^{2}-(\alpha+\beta) x +\alpha \beta=0\) i.e. \(x ^{2}-\) (sum of roots) \(x +\) product of roots \(=0\)
(e) If \(\alpha, \beta\) are roots of equation \(a x^{2}+b x+c=0\), we have identity in \(x\) as \(a x^{2}+b x+c=a(x-\alpha)(x-\beta)\)
2. NATURE OF ROOTS :
(a) Consider the quadratic equation \(a x^{2}+b x+c=0\) where \(a, b\), \(c \in R \,\, \&\,\, a \neq 0\) then \(;\)
- \(D >0 \Leftrightarrow\) roots are real & distinct (unequal).
- \(D =0 \Leftrightarrow\) roots are real & coincident (equal)
- \(D <0 \Leftrightarrow\) roots are imaginary.
- If \(p + i q\) is one root of a quadratic equation, then the other root must be the conjugate \(p - i q\) & vice versa. \((p, q \in R \) & \(i=\sqrt{-1})\)
(b) Consider the quadratic equation \(a x^{2}+b x+c=0\) where \(a , b , c \in Q \,\, \&\,\, a \neq 0\) then
- If \(D\) is a perfect square, then roots are rational.
- If \(\alpha= p +\sqrt{ q }\) is one root in this case, ( where \(p\) is rational \(\& \sqrt{ q }\) is a surd) then other root will be \(p -\sqrt{ q }\).
3. COMMON ROOTS OF TWO QUADRATIC EQUATIONS
(a) Atleast one common root.
Let \(\alpha\) be the common root of \(a x^{2}+b x+c=0\) & \(a^{\prime} x^{2}+b^{\prime} x+c^{\prime}=0\)
then \(a \alpha^{2}+b \alpha+c=0 \& a^{\prime} \alpha^{2}+b^{\prime} \alpha+c^{\prime}=0 .\) By Cramer's
Rule \(\dfrac{\alpha^{2}}{b c^{\prime}-b^{\prime} c}=\dfrac{\alpha}{a^{\prime} c-a c^{\prime}}=\dfrac{1}{a b^{\prime}-a^{\prime} b}\)
Therefore, \(\alpha=\dfrac{c a^{\prime}-c^{\prime} a}{a b^{\prime}-a^{\prime} b}=\dfrac{b c^{\prime}-b^{\prime} c}{a^{\prime} c-a c^{\prime}}\)
So the condition for a common root is \(\left(c a^{\prime}-c^{\prime} a\right)^{2}=\left(a b^{\prime}-a^{\prime} b\right)\left(b c^{\prime}-b^{\prime} c\right)\)
(b) If both roots are same then \(\dfrac{ a }{ a ^{\prime}}=\dfrac{ b }{ b ^{\prime}}=\dfrac{ c }{ c ^{\prime}}\)
4. ROOTS UNDER PARTICULAR CASES
Let the quadratic equation \(a x^{2}+b x+c=0\) has real roots and
(a) If \(b =0 \Rightarrow\) roots are of equal magnitude but of opposite sign.
(b) If \(c=0 \Rightarrow\) one roots is zero other is \(-b / a\).
(c) If \(a = c \Rightarrow\) roots are reciprocal to each other.
(d) If \(a c<0 \Rightarrow\) roots are of opposite signs.
(e) If \(\left.\begin{array}{r} a >0, b >0, c >0 \\ a <0, b <0, c <0\end{array}\right\} \Rightarrow\) both roots are negative.
(f) If \(\left.\begin{array}{r}a>0, b<0, c>0 \\ a<0, b>0, c<0\end{array}\right\} \Rightarrow\) both roots are positive.
(g) If sign of \(a=\) sign of \(b \neq\) sign of \(c\) \(\Rightarrow\) Greater root in magnitude is negative.
(h) If sign of \(b=\) sign of \(c \neq\) sign of a
\(\Rightarrow\) Greater root in magnitude is positive.
(i) If \(a+b+c=0 \Rightarrow\) one root is 1 and second root is \(c / a\).
5. MAXIMUM & MINIMUM VALUES OF QUADRATIC EXPRESSION:
Maximum or Minimum Values of expression \(y=a x^{2}+b x+c\) is \(\dfrac{- D }{4 a }\) which occurs at \(x =-( b / 2 a )\) according as \(a <0\) or \(a >0 .\)
\(y \in\left[\dfrac{-D}{4 a}, \infty\right)\) if \(a>0 \quad \& \quad y \in\left(-\infty, \frac{-D}{4 a}\right]\) if \(a<0 .\)
6. LOCATION OF ROOTS :
Let \(f ( x )= ax ^{2}+ bx + c\), where \(a , b , c \in R , a \neq 0\)
(a) Conditions for both the roots of \(f ( x )=0\) to be greater than a specified number 'd' are \(\mathbf{D \geq 0}\) and \(\mathbf {a . f ( d )> 0 \,\, \& \,\, ( - b / 2 a )> d}\).
(c) Condition for exactly one root of \(f ( x )=0\) to lie in the interval \(( d , e)\) i.e. \(d < x < e\) is \(\mathbf{f ( d ) \cdot f ( e )< 0}\)
(d) Conditions that both roots of \(f(x)=0\) to be confined between the numbers d & e are (here \(d < e ).\)
\(\mathbf{D \geq 0}\) and \(\mathbf{a \cdot f ( d )> 0\,\, \& \,\, a f ( e )> 0}\) and \(\mathbf{d <(- b / 2 a )< e}\)
7. GENERAL QUADRATIC EXPRESSION IN TWO VARIABLES :
\(f(x, y)=a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c\) may be resolved into two linear factors if ;
\(\triangle = abc + 2fgh – af^2 – bg^2– ch^2= 0\) OR \(\left|\begin{array}{lll}a & h & g \\h & b & f \\g & f & c\end{array}\right|=0\)
8. THEORY OF EQUATIONS :
If \(\alpha_{1}, \alpha_{2}, \alpha_{3}, \ldots \ldots \ldots \alpha_{n}\) are the roots of the equation;
\(f(x)=a_{0} x^{n}+a_{1} x^{n-1}+a_{2} x^{n-2}+\ldots . .+a_{n-1} x+a_{n}=0\)
where \(a _{0}, a _{1}, \ldots \ldots a _{ n }\) are constants \(a _{0} \neq 0\) then,
\(\begin{aligned}\sum \alpha_{1} &=-\frac{ a _{1}}{ a _{0}}, \, \sum\alpha_{1} \alpha_{2}=+\frac{ a _{2}}{ a _{0}},\, \sum \alpha_{1} \alpha_{2} \alpha_{3} \\&=-\frac{ a _{3}}{ a _{0}}, \ldots \ldots, \alpha_{1}\alpha_{2} \alpha_{3} \ldots \ldots \alpha_{ n }(-1)^{ n } \frac{ a _{ n }}{ a _{0}}\end{aligned}\)
Note:
(i) Every odd degree equation has at least one real root whose sign is opposite to that of its constant term, when coefficient of highest degree term is (+)ve {if not then make it (+)ve}.
\(\text { Ex.} \,\, x^{3}-x^{2}+x-1=0\)
(ii) Even degree polynomial whose constant term is (-)ve & coefficient of highest degree term is (+)ve has atleast two real roots, one (+)ve & one (-)ve.
(iii) If equation contains only even power of x & all coefficient are (+)ve, then all roots are imaginary.
(iv) Rational root theorem : If a rational number \(\dfrac{ p }{ q }\left( p , q \in Z _{0}\right)\) is a root of polynomial equation with integral coefficient \(a_{n} x^{n}+a_{n-1} x^{n-1}+\ldots \ldots+a_{0}=0\), then \(p\) divides \(a_{0}\) and \(q\) divides \(a _{ n }\).
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