**Question:**

Write the discriminant of the following quadratic equations:

(i) $2 x^{2}-5 x+3=0$

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

(iii) $(x-1)(2 x-1)=0$

(iv) $x^{2}-2 x+k=0, k \in \mathrm{R}$

(v) $\sqrt{3} x^{2}+2 \sqrt{2} x-2 \sqrt{3}=0$

(vi) $x^{2}-x+1=0$

**Solution:**

We have to find the discriminant of the following quadratic equations

(i) We have been given, $2 x^{2}-5 x+3=0$

Now we also know that for an equation $a x^{2}+b x+c=0$, the discriminant is given by the following equation:

$D=b^{2}-4 a c$

Now, according to the equation given to us, we have, $a=2, b=-5$ and $c=3$.

Therefore, the discriminant is given as,

$D=(-5)^{2}-4(2)(3)$

$=25-24$

$=1$

Therefore, the discriminant of the equation is 1 .

(ii) We have been given, $x^{2}+2 x+4=0$

Now we also know that for an equation $a x^{2}+b x+c=0$, the discriminant is given by the following equation:

$D=b^{2}-4 a c$

Now, according to the equation given to us, we have, $a=1, b=2$ and $c=4$.

Therefore, the discriminant is given as,

$D=(2)^{2}-4(1)(4)$

$=4-16$

$=-12$

Therefore, the discriminant of the equation is $-12$.

(iii) We have been given, $(x-1)(2 x-1)=0$

Now, simplify the equation to be represented in the quadratic form, so we have

$2 x^{2}-x-2 x+1=0$

$2 x^{2}-3 x+1=0$

Now we also know that for an equation $a x^{2}+b x+c=0$, the discriminant is given by the following equation:

$D=b^{2}-4 a c$

Now, according to the equation given to us, we have, $a=2, b=-3$ and $c=1$.

Therefore, the discriminant is given as,

$D=(-3)^{2}-4(2)(1)$

$=9-8$

$=1$

Therefore, the discriminant of the equation is 1 .

(iv) We have been given, $x^{2}-2 x+k=0$

$D=b^{2}-4 a c$

Now, according to the equation given to us, we have, $a=1, b=-2$ and $c=k$.

Therefore, the discriminant is given as,

$D=(-2)^{2}-4(1)(k)$

$=4-4 k$

Therefore, the discriminant of the equation is $4-4 k$.

(v) We have been given, $\sqrt{3} x^{2}+2 \sqrt{2} x-2 \sqrt{3}=0$

$D=b^{2}-4 a c$

Now, according to the equation given to us, we have, $a=\sqrt{3}, b=2 \sqrt{2}$ and $c=-2 \sqrt{3}$.

Therefore, the discriminant is given as,

$D=(2 \sqrt{2})^{2}-4(\sqrt{3})(-2 \sqrt{3})$

$=8+24$

$=32$

Therefore, the discriminant of the equation is 32 .

(vi) We have been given, $x^{2}-x+1=0$

$D=b^{2}-4 a c$

Now, according to the equation given to us, we have, $a=1, b=-1$ and $c=1$.

Therefore, the discriminant is given as,

$D=(-1)^{2}-4(1)(1)$

$=1-4$

$=-3$

Therefore, the discriminant of the equation is $-3$.