Name: Condition for Common Roots of Quadratic Equation
Uploaded: 2019-07-19
Duration: 36 min 41 s
Description: In this video, we will learn the condition for one common root and the condition for both roots common for two given quadratic equations with the help of exa...

Question

Find the set of values of

' a'

if

{(x^{2} + x)}^{2} + a (x^{2} + x) + 4 = 0

has

(i)

All four real & distinct roots.

(i i)

Four roots in which only two roots are real and distinct.

(i i i)

All four imaginary roots.

(i v)

Four real roots in which only two are equal.

Accepted Answer

${(x^{2} + x)}^{2} + a (x^{2} + x) + 4 = 0$

Let $x^{2} + x = t$ .

Also, minimum value of $x^{2} + x$ occurs at $x = - \frac{b}{2 a} = - \frac{1}{2}$ and is equal to ${(- \frac{1}{2})}^{2} + (- \frac{1}{2}) = - \frac{1}{4}$ .

$\Rightarrow t = x^{2} + x$ $\Rightarrow t \in [- \frac{1}{4}, \infty) \forall x \in R$

Given equation become $t^{2} + a t + 4 = 0$ .

$(i)$ All four real & distinct roots.

Both roots of quadratic equation $t^{2} + a t + 4 = 0$ should be greater than $- \frac{1}{4}$ and distinct.

So, we need to apply three conditions $D > 0, a \cdot f (d) > 0 & - \frac{b}{2 a} > d$ .

$(1)$ $D > 0$

$\Rightarrow {(a)}^{2} - 4 (4) > 0$

$\Rightarrow {(a)}^{2} > 16$

$\Rightarrow a \in (- \infty, - 4) \cup (4, \infty) . . . (i)$

$(2)$ $a . f (d) > 0 \Rightarrow 1 . f (- \frac{1}{4}) > 0$

$\Rightarrow {(- \frac{1}{4})}^{2} + a (- \frac{1}{4}) + 4 > 0$

$\Rightarrow \frac{1}{16} - \frac{a}{4} + 4 > 0$

$\Rightarrow a < \frac{65}{4}$

$\Rightarrow a \in (- \infty, \frac{65}{4}) . . . (ii)$

$(3)$ $- \frac{b}{2 a} > d$

$\Rightarrow - \frac{a}{2} > - \frac{1}{4}$

$\Rightarrow a < \frac{1}{2}$

$\Rightarrow a \in (- \infty, \frac{1}{2}) . . . (iii)$

Intersection of $(i), (ii)$ and $(iii)$ will be $\Rightarrow a \in (- \infty, - 4)$

$(i i)$ Four roots in which only two roots are real and distinct.

One roots of quadratic equation $t^{2} + a t + 4 = 0$ is greater then $- \frac{1}{4}$ and the other root is smaller than $- \frac{1}{4}$ .

So, $a \cdot f (d) < 0$ is necessary and sufficient condition.

$a . f (d) < 0 \Rightarrow 1 \cdot f (- \frac{1}{4}) < 0$

$\Rightarrow {(- \frac{1}{4})}^{2} + a (- \frac{1}{4}) + 4 < 0$

$\Rightarrow \frac{1}{16} - \frac{a}{4} + 4 < 0$

$\Rightarrow a > \frac{65}{4}$

$\Rightarrow a \in (\frac{65}{4}, \infty)$

$(i i i)$ All four imaginary roots.

Case $1$ - Both roots of quadratic equation $t^{2} + a t + 4 = 0$ are imaginary.

$\Rightarrow D < 0$

$\Rightarrow {(a)}^{2} - 4 (4) < 0 \Rightarrow a^{2} - 16 < 0$

$\Rightarrow a^{2} < 16 \Rightarrow a \in (- 4, 4) . . . (iv)$

Case $2$ - Both roots of quadratic equation $t^{2} + a t + 4 = 0$ are less than $- \frac{1}{4}$ .

So, we need to apply three conditions $D \geq 0, a \cdot f (d) > 0 & - \frac{b}{2 a} < d$

$(1)$ $D \geq 0$

$\Rightarrow {(a)}^{2} - 4 (4) \geq 0$

$\Rightarrow {(a)}^{2} \geq 16$

$\Rightarrow a \in (- \infty, - 4] \cup [4, \infty) . . . (v)$

$(2)$ $a . f (d) > 0 \Rightarrow 1 . f (- \frac{1}{4}) > 0$

$\Rightarrow {(- \frac{1}{4})}^{2} + a (- \frac{1}{4}) + 4 > 0$

$\Rightarrow \frac{1}{16} - \frac{a}{4} + 4 > 0$

$\Rightarrow a < \frac{65}{4}$

$\Rightarrow a \in (- \infty, \frac{65}{4}) . . . (vi)$

$(3)$ $- \frac{b}{2 a} < d$

$\Rightarrow - \frac{a}{2} < - \frac{1}{4}$

$\Rightarrow a > \frac{1}{2}$

$\Rightarrow a \in (\frac{1}{2}, \infty) . . . (vii)$

Intersection of $(v), (vi)$ and $(vii)$ will be $\Rightarrow a \in [4, \frac{65}{4}) . . . (viii)$

Union of Case $1$ and Case $2$ $\Rightarrow a \in (- 4, \frac{65}{4})$

$(i v)$ Four real roots in which only two are equal.

One root of quadratic equation $t^{2} + a t + 4 = 0$ is greater than $- \frac{1}{4}$ and one root is equal to $- \frac{1}{4}$ .

So, we need to apply three conditions $D > 0, f (- \frac{1}{4}) = 0 & - \frac{b}{2 a} > d$

$(1)$ $D > 0$

$\Rightarrow {(a)}^{2} - 4 (4) > 0$

$\Rightarrow {(a)}^{2} > 16$

$\Rightarrow a \in (- \infty, - 4) \cup (4, \infty) . . . (ix)$

$(2)$ $f (- \frac{1}{4}) = 0$

$\Rightarrow {(- \frac{1}{4})}^{2} + a (- \frac{1}{4}) + 4 = 0$

$\Rightarrow \frac{1}{16} - \frac{a}{4} + 4 = 0$

$\Rightarrow a = \frac{65}{4}$

$\Rightarrow a \in \{\frac{65}{4}\} . . . (x)$

$(3)$ $- \frac{b}{2 a} > d$

$\Rightarrow - \frac{a}{2} > - \frac{1}{4}$

$\Rightarrow a < \frac{1}{2}$

$\Rightarrow a \in (- \infty, \frac{1}{2}) . . . (xi)$

Intersection of $(ix), (x)$ and $(xi)$ will be $\Rightarrow a \in ϕ$

Find the set of values of $' a'$ if ${(x^{2} + x)}^{2} + a (x^{2} + x) + 4 = 0$ has
$(i)$ All four real & distinct roots.
$(i i)$ Four roots in which only two roots are real and distinct.
$(i i i)$ All four imaginary roots.
$(i v)$ Four real roots in which only two are equal.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Find the set of values of 'a' if x2+x2+ax2+x+4=0 has i All four real & distinct roots. ii Four roots in which only two roots are real and distinct. iii All four imaginary roots. iv Four real roots in which only two are equal.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

Find the set of values of $' a'$ if ${(x^{2} + x)}^{2} + a (x^{2} + x) + 4 = 0$ has
$(i)$ All four real & distinct roots.
$(i i)$ Four roots in which only two roots are real and distinct.
$(i i i)$ All four imaginary roots.
$(i v)$ Four real roots in which only two are equal.