Question

Find the nature of the roots of the following quadratic equation:

1.$3x^2+7x+2=0$

2.${\left(x+3\right)}^2+4x+7=0$

3.$2x^2-8x+34=0$

4.$x^2+2\sqrt{3x}-1=0$

Answer 1

A. $3x^2+7x+2=0$

$△=b^2-4ac=7^2-4\left(2\right)\left(3\right)=49-24=25>0$

$\therefore$Roots are real and positive

B. ${(x+3)}^2+4x+7=0$

$x^2+9+6x+4x+7=0$

$x^2+10x+16=0$

$△=b^2-4ac=100-4\left(16\right)=36=6^2>0$

$\therefore$Roots are real and positive

C. $2x^2-8x+34=0$

$△=8^2-4\left(2\right)\left(34\right)<0$

$\therefore$ Roots are imaginary

D. $x^2+2\sqrt{3}x-1=0$

$△={\left(2\sqrt{3}\right)}^2+4\left(1\right)=12+4=16>0$

Roots are real and positive.