Question

Match the following quadratic equations with the nature of roots

• A. No real roots
• B. $\mathrm{Rational}\mathrm{and}\mathrm{distinct}\mathrm{roots}$
• C. $\mathrm{Distinct}\mathrm{and}\mathrm{integer}\mathrm{roots}$
• D. $\mathrm{Irrational}\mathrm{and}\mathrm{unequal}\mathrm{roots}$

Answer 1

$a)x^2-4x+3$
On comparing with the standard quadratic equation $a{x}^2+bx+c=0$ we get, $a=1,b=-4,c=3.$
We know $D=b^2-4ac=(-4{)}^2-\mathrm{4.1.3}$
$\Rightarrow D=4$
$\Rightarrow D$ is a perfect square of a integer number
$\therefore$ real and distinct integer roots

$b)3x^2-\frac{11}{2}x+\frac{5}{2}$
On comparing with the standard quadratic equation $a{x}^2+bx+c=0$ we get, $a=3,b=-\frac{11}{2},c=\frac{5}{2}.$
We know $D=b^2-4ac=(-\frac{11}{2}{)}^2-\mathrm{4.3.}\frac{5}{2}$
$\Rightarrow D=\frac{1}{2}$
$\Rightarrow D$ is a perfect square of a rational number
$\therefore$ real and distinct rational roots

$c)2x^2+4x-5$
On comparing with the standard quadratic equation $a{x}^2+bx+c=0$ we get, $a=2,b=4,c=-5.$
We know $D=b^2-4ac=(4{)}^2-\mathrm{4.2.}(-5)$
$\Rightarrow D=56$
$\Rightarrow D$ is not a perfect square
$\therefore$ real and distinct integer roots

$d)x^2-2x+7$
On comparing with the standard quadratic equation $a{x}^2+bx+c=0$ we get, $a=1,b=-2,c=7.$
We know $D=b^2-4ac=(-2{)}^2-\mathrm{4.1.}(7)$
$\Rightarrow D=-24$
$\Rightarrow D$ is less than zero
$\therefore$ no real roots