Quadratic Equations and Expressions
6.0 Location of roots
6.0 Location of roots
Let $f(x) = a{x^2} + bx + c\quad a,b,c \in R{\text{ and }}a \ne 0$
$\alpha ,\beta $ be the roots of $f(x)=0$ and $k,{k_1},{k_2} \in R\quad {k_1} < {k_2}$.
| S.No | Conditions | Result |
| 1. | If both the roots are positive i.e., they lie in $(0,\infty )$ | $$D \geqslant 0{\text{ i}}{\text{.e}}{\text{., }}{b^2} - 4ac \geqslant 0$$ Sum of the roots as well as product of the roots must be positive i.e., $$\alpha + \beta = - \frac{b}{a} > 0{\text{ and }}\alpha \beta = \frac{c}{a} > 0$$ |
| 2. | If both the roots are negative i.e., they lie in $( - \infty ,0)$ | $$D \geqslant 0{\text{ i}}{\text{.e}}{\text{., }}{b^2} - 4ac \geqslant 0$$ Sum of the roots must be negative and product of the roots must be positive i.e., $$\alpha + \beta = - \frac{b}{a} < 0{\text{ and }}\alpha \beta = \frac{c}{a} > 0$$ |
| 3. | Condition for a number $k$ if both the roots are less than $k$ | 
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| 4. | Condition for a number $k$ if both the roots are greater than $k$ | 
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| 5. | Condition for a number $k$ if $k$ lies between the roots | 
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| 6. | Condition for numbers ${k_1}$ and ${k_2}$ if exactly one root lies in the interval $({k_1},{k_2})$ | 
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| 7. | Condition for numbers ${k_1}$ and ${k_2}$ if both roots are confined between ${k_1}$ and ${k_2}$ | 
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| 8. | Condition for numbers ${k_1}$ and ${k_2}$ if ${k_1}$ and ${k_2}$ lie between the roots | 
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Question 6. If $\alpha $ is the root of the equation $a{x^2} + bx + c = 0$ and $\beta $ is a root of the equation $ - a{x^2} + bx + c = 0$, then prove that there will be a root of the equation $\frac{a}{2}{x^2} + bx + c = 0$ lying between $\alpha $ and $\beta $.
Solution: Let $f(x) = \frac{a}{2}{x^2} + bx + c$ $$\begin{equation} \begin{aligned} f(\alpha ) = \frac{a}{2}{\alpha ^2} + b\alpha + c \\ {\text{ }} = a{\alpha ^2} + b\alpha + c - \frac{a}{2}{\alpha ^2} \\ {\text{ = }} - \frac{a}{2}{\alpha ^2} \\\end{aligned} \end{equation} $$ As $\alpha $ is a root of $a{x^2} + bx + c = 0$. Also $$\begin{equation} \begin{aligned} f(\beta ) = \frac{a}{2}{\beta ^2} + b\beta + c \\ {\text{ }} = - a{\beta ^2} + b\beta + c + \frac{3}{2}a{\beta ^2} \\ {\text{ = }}\frac{3}{2}a{\beta ^2} \\\end{aligned} \end{equation} $$ As $\beta $ is a root of the equation $ - a{x^2} + bx + c = 0$.
Now, $$f(\alpha ).f(\beta ) = - \frac{3}{4}{a^2}{\alpha ^2}{\beta ^2} < 0$$
$ \Rightarrow f(x) = 0$ has one real root between $\alpha $ and $\beta $.
