Question

Determine the nature of the roots of the following quadratic equations:

(i) 2x² − 3x + 5 = 0
(ii) 2x² − 6x + 3 = 0
(iii) $\frac{3}{5}{x}^2-\frac{2}{3}x+1=0$
(iv) $3x^2-4\sqrt{3}x+4=0$
(v) $3x^2-2\sqrt{6}x+2=0$
(vi) $4x^2+4\sqrt{3}x+3=0$

Answer 1

(i) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(ii) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(iii) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(iv) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(v) The given quadric equation is

Here,

As we know that

Putting the value of

Since,

Therefore, root of the given equation are.

(vi) $4x^2+4\sqrt{3}x+3=0$

$$\begin{gathered} 4x^2+4\sqrt{3}x+3=0 \\ \mathrm{Discriminant}=b^2-4ac \\ ={\left(4\sqrt{3}\right)}^2-4\left(4\right)\left(3\right) \\ =48-48 \\ =0 \end{gathered}$$

Since, Discriminant = 0
Therefore, the roots are real and equal.

Hence, the given equation has real and equal roots.