Question

Determine the nature of the roots of the following quadratic equations;
$\left(i\right)(x-2a)(x-2b)=4ab\left(ii\right)9a^2{b}^2{x}^2-24abcdx+16c^2{d}^2=0,a\ne 0,b\ne 0\left(iii\right)2(a^2+b^2)x^2+2(a+b)x+1=0\left(iv\right)(b+c)x^2-(a+b+c)x+a=0$

Answer 1

For an equation the nature of the roots can be determined by the value of D.
where D =

If,
D > 0, real and unequal roots
D=0 , real and equal roots
D<0 , real roots dont exist.

(i)


D =
As D > 0, roots are real and distinct.

(ii)

D = -
D = 0
Hence, roots are real and equal.

(iii)

$D=\left(2\right(a+b){)}^2-4\times 2(a^2+b^2)\times 1$

$=4(a^2+b^2+2ab)-8(a^2+b^2)$

$=-4(a^2+b^2-2ab)$

$=-4(a-b{)}^2$

As D < 0, roots are not real.

(iv)

D =

Hence, roots are real.