Differentiate the following functions with respect to x:
For given y, prove the following
For given y, prove the following
For given y, prove the following
For given y, prove the following
For given y, prove the following
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
For given y prove that
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
Differentiate the following functions with respect to x:
If the roots of the equations and are simultaneously real then prove that
It is given that the roots of the equation $a x^{2}+2 b x+c=0$ are real. $\begin{array}{l} \therefore D_{1}=(2 b)^{2}-4 \times a \times c \geq 0 \\ \Rightarrow 4\left(b^{2}-a c\right) \geq 0 \\...
If the roots of the equation are equal, prove that
It is given that the roots of the equation $\left(a^{2}+b^{2}\right) x^{2}-2(a c+b d) x+\left(c^{2}+d^{2}\right)=0$ are equal. $\begin{array}{l} \therefore D=0 \\ \Rightarrow[-2(a c+b...
If a and are real and then show that the roots of the equation are equal and unequal.
The given equation is $(a-b) x^{2}+5(a+b) x-2(a-b)=0$. $\begin{array}{l} \therefore D=[5(a+b)]^{2}-4 \times(a-b) \times[-2(a-b)] \\ =25(a+b)^{2}+8(a-b)^{2} \end{array}$ Since a and $\mathrm{b}$ are...
Find the values of for which the given quadratic equation has real and distinct roots:
(i) .
(ii) .
(i) The given equation is $9 x^{2}+3 k x+4=0$. $\therefore D=(3 k)^{2}-4 \times 9 \times 4=9 k^{2}-144$ The given equation has real and distinct roots if $D>0 .$ $\begin{array}{l} \therefore 9...
Find the values of for which the given quadratic equation has real and distinct roots:
(i) .
(ii) .
(i) The given equation is $k x^{2}+6 x+1=0$. $\therefore D=6^{2}-4 \times k \times 1=36-4 k$ The given equation has real and distinct roots if $D>0$. $\begin{array}{l} \therefore 36-4 k>0 \\...
Find the value of for which the roots of are real and equal
Given: $\begin{array}{l} 9 x^{2}+8 k x+16=0 \\ \text { Here, } \\ a=9, b=8 k \text { and } c=16 \end{array}$ It is given that the roots of the equation are real and equal; therefore, we have:...
Find the value of a for which the equation has equal roots.
$(\alpha-12) x^{2}+2(\alpha-12) x+2=0$ Here, $a=(\alpha=12), b=2(\alpha-12) \text { and } c=2$ It is given that the roots of the equation are equal; therefore, we have $\begin{array}{l} D=0 \\...
Find the value of for which the quadratic equation has real roots.
$2 x^{2}+p x+8=0$ Here, $a=2, b=p$ and $c=8$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =p^{2}-4 \times 2 \times 8 \\ =\left(p^{2}-64\right) \end{array}$ If $D...
If the roots of the quadratic equation are real and equal, show that either or
$\left(c^{2}-a b\right) x^{2}-2\left(a^{2}-b c\right) x+\left(b^{2}-a c\right)=0$ Here, $a=\left(c^{2}-a b\right), b=-2\left(a^{2}-b c\right), c=\left(b^{2}-a c\right)$ It is given that the roots of...
If the quadratic equation has equal roots, prove that
$\left(1+m^{2}\right) x^{2}+2 m c x+\left(c^{2}-a^{2}\right)=0$ Here, $a=\left(1+m^{2}\right), b=2 m c \text { and } c=\left(c^{2}-a^{2}\right)$ It is given that the roots of the equation are equal;...
If is a root of the equation . find the value of for the which the quadratic equation has equal roots.
It is given that $-4$ is a root of the quadratic equation $x^{2}+2 x+4 p=0$ $\begin{array}{l} \therefore(-4)^{2}+2 \times(-4)+4 p=0 \\ \Rightarrow 16-8+4 p=0 \\ \Rightarrow 4 p+8=0 \\ \Rightarrow...
If 3 is a root of the quadratic equation ., find the value of so that the roots of the equation are equal.
It is given that 3 is a root of the quadratic equation $x^{2}-x+k=0$. $\begin{array}{l} \therefore(3)^{2}-3+k=0 \\ \Rightarrow k+6=0 \\ \Rightarrow k=-6 \end{array}$ The roots of the equation...
If is a root of the quadratic equation . and the quadratic equation has equal roots, find the value of .
It is given that $-5$ is a root of the quadratic equation $2 x^{2}+p x- 15=0$ $\therefore 2(-5)^{2}+p \times(-5)-15=0$ $\begin{array}{l} \Rightarrow-5 p+35=0 \\ \Rightarrow p=7 \end{array}$ The...
Find the values of for which the quadratic equation . , has equal roots. Hence find the roots of the equation.
The given equation is $(p+1) x^{2}-6(p+1) x+3(p+9)=0$. This is of the form $a x^{2}+b x+c=0$, where $a=p+1, b=-6(p+1)$ and $c=3(p+9)$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =[-6(p+1)]^{2}-4...
Find the value of for which the quadratic equation . has real and equal roots.
The given equation is $(2 p+1) x^{2}-(7 p+2) x+(7 p-3)=0$. This is of the form $a x^{2}+b x+c=0$, where $a=2 p+1, b=-(7 p+2)$ and $c=7 p-3$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =-[-(7...
Find the values of for which the quadratic equation . has real and equal roots.
The given equation is $(3 k+1) x^{2}+2(k+1) x+1=0$. This is of the form $a x^{2}+b x+c=0$, where $a=3 k+1, b=2(k+1)$ and $c=1$. $\begin{array}{l} \therefore D=b^{2}-4 a c \\ =[2(k+1)]^{2}-4 \times(3...
Find the nonzero value of for which the roots of the quadratic equation . are real and equal.
The given equation is $9 x^{2}-3 k x+k=0$. This is of the form $a x^{2}+b x+c=0$, where $a=9, b=-3 k$ and $c=k$. $\therefore D=b^{2}-4 a c=(-3 k)^{2}-4 \times 9 \times k=9 k^{2}-36 k$ The given...
For what values of are the roots of the equation . real and equal?
The given equation is $4 x^{2}+p x+3=0$. This is of the form $a x^{2}+b x+c=0$, where $a=4, b=p$ and $c=3$. $\therefore D=b^{2}-4 a c=p^{2}-4 \times 4 \times 3=p^{2}-48$ The given equation will have...
For what value of are the roots of the quadratic equation real and equal.
The given equation is $\begin{array}{l} k x(x-2 \sqrt{5})+10=0 \\ \Rightarrow k x^{2}-2 \sqrt{5} k x+10=0 \end{array}$ This is of the form $a x^{2}+b x+c=0$, where $a=k, b=-2 \sqrt{5} k$ and $c=10$....
For what values of are the roots of the quadratic equation real and equal?
Given: $3 x^{2}+2 k x+27=0$ Here, $a=3, b=2 k \text { and } c=27$ It is given that the roots of the equation are real and equal; therefore, we have: $\begin{array}{l} D=0 \\ \Rightarrow(2 k)^{2}-4...
Show that the roots of the equation are real for all real values of and .
$x^{2}+p x-q^{2}=0$ Here, $a=1, b=p$ and $c=-q^{2}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =p^{2}-4 \times 1 \times\left(-q^{2}\right) \\ =\left(p^{2}+4...
If a and b are distinct real numbers, show that the quadratic equations has no real roots.
The given equation is $2\left(a^{2}+b^{2}\right) x^{2}+2(a+b) x+1=0$ $\begin{array}{l} \therefore D=[2(a+b)]^{2}-4 \times 2\left(a^{2}+b^{2}\right) \times 1 \\ =4\left(a^{2}+2 a...
Find the nature of roots of the following quadratic equations:
(i)
(ii) .
(i) The given equation is $12 x^{2}-4 \sqrt{15} x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=12, b=-4 \sqrt{15}$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-4 \sqrt{15})^{2}-4...
Find the nature of roots of the following quadratic equations:
(i)
(ii)
(i) The given equation is $5 x^{2}-4 x+1=0$ This is of the form $a x^{2}+b x+c=0$, where $a=5, b=-4$ and $c=1$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-4)^{2}-4 \times 5 \times 1=16-20=-4<0$...
Find the nature of roots of the following quadratic equations:
(i)
(ii)
(i) The given equation is $2 x^{2}-8 x+5=0$ This is of the form $a x^{2}+b x+c=0$, where $a=2, b=-8$ and $c=5$. $\therefore$ Discriminant, $D=b^{2}-4 a c=(-8)^{2}-4 \times 2 \times 5=64-40=24>0$...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: , where and
$12 a b x^{2}-\left(9 a^{2}-8 b^{2}\right) x-6 a b=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=12 a b, B=-\left(9 a^{2}-8 b^{2}\right)$ and $C=-6 a b$ Discriminant $D$ is given by:...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: and
The given equation is $a^{2} b^{2} x^{2}-\left(4 b^{4}-3 a^{4}\right) x-12 a^{2} b^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=a^{2} b^{2}, B=-\left(4 b^{4}-3 a^{4}\right)$ and $c=-12...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
$3 a^{2} x^{2}+8 a b x+4 b^{2}=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=3 a^{2}, B=8 a b$ and $C=4 b^{2}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(B^{2}-4 A C\right) \\...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $x^{2}-(2 b-1) x+\left(b^{2}-b-20\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-(2 b-1)$ and $C=b^{2}-b-20$ $\therefore$ Discriminant, $D=B^{2}-4 A C=[-(2...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $4 x^{2}-4 b x-\left(a^{2}-b^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=4 b$ and $C=-\left(a^{2}-b^{2}\right)$ $\therefore$ Discriminant, $D=B^{2}-4 A...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $4 x^{2}-4 a^{2} x+\left(a^{4}-b^{4}\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=4, B=-4 a^{2}$ and $C=a^{4}-b^{4}$ $\therefore$ Discriminant, $B^{2}-4 A...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $x^{2}-4 a x-b^{2}+4 a^{2}=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-4 a$ and $C=-b^{2}+4 a^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-4 a)^{2}-4 \times 1...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $x^{2}+5 x-\left(a^{2}+a-6\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=5$ and $C=-\left(a^{2}+a-8\right)$ $\therefore$ Discriminant, $D=$ $\begin{array}{l}...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $x^{2}+6 x-\left(a^{2}+2 a-8\right)=0$. Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=6$ and $C=-\left(a^{2}+2 a-8\right)$ $\therefore$ Discriminant, $D=$...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $x^{2}-2 a x-\left(4 b^{2}-a^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=1, B=-2 a$ and $C=-\left(4 b^{2}-a^{2}\right)$ $\therefore$ Discriminant, $B^{2}-4 A...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $x^{2}-2 a x+\left(a^{2}-b^{2}\right)=0$ On comparing it with $A x^{2}+B x+C=0$, we get: $A=1, B=-2 a$ and $C=\left(a^{2}-b^{2}\right)$ Discriminant $D$ is given by: $\begin{array}{l}...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $36 x^{2}-12 a x+\left(a^{2}-b^{2}\right)=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=36, B=-12 a$ and $C=a^{2}-b^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=(-12...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $\begin{array}{l} \frac{m}{n} x^{2} \frac{n}{m}=1-2 x \\ \Rightarrow \frac{m^{2} x^{2}+n^{2}}{m n}=1-2 x \\ \Rightarrow m^{2} x^{2}+n^{2}=m n-2 m n x \\ \Rightarrow m^{2}...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $\begin{array}{l} x-\frac{1}{x}=3, x \neq 0 \\ \Rightarrow \frac{x^{2}-1}{x}=3 \\ \Rightarrow x^{2}-1=3 x \\ \Rightarrow x^{2}-3 x-1=0 \end{array}$ This equation is of the form...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $\begin{array}{l} \frac{1}{x}-\frac{1}{x-2}=3, x \neq 0,2 \\ \Rightarrow \frac{x-2-x}{x(x-2)}=3 \\ \Rightarrow \frac{-2}{x^{2}-2 x}=3 \\ \Rightarrow-2=3 x^{2}-6 x \\...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $\begin{array}{l} x+\frac{1}{x}=3, x \neq 0 \\ \Rightarrow \frac{x^{2}+1}{x}=3 \\ \Rightarrow x^{2}+1=3 x \\ \Rightarrow x^{2}-3 x+1=0 \end{array}$ This equation is of the form...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $3 x^{2}-2 x+2=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=(-2)^{2}-4 \times 3 \times 2=4-24=-20<0$ Hence,...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $2 x^{2}+5 \sqrt{3} x+6=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=5 \sqrt{3}$ and $c=6$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(5 \sqrt{3})^{2}-4 \times 2...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $x^{2}-(\sqrt{3}+1) x+\sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=-(\sqrt{3}+1)$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $2 x^{2}+a x-a^{2}=0$ Comparing it with $A x^{2}+B x+C=0$, we get $A=2, B=a$ and $C=-a^{2}$ $\therefore$ Discriminant, $D=B^{2}-4 A C=a^{2}-4 \times 2 \times-a^{2}=a^{2}+8...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $x^{2}+x+2=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=1, b=1$ and $c=2$ $\therefore$ Discriminant $D=b^{2}-4 a c=1^{2}-4 \times 1 \times 2=1-8=-7<0$ Hence, the given...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $2 \sqrt{3} x^{2}-5 x+\sqrt{3}=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2 \sqrt{3}, b=-5$ and $c=\sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-5)^{2}-4 \times...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $3 x^{2}-2 \sqrt{6} x+2=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=3, b=-2 \sqrt{6}$ and $c=2$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{6})^{2}-4 \times 3...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $4 \sqrt{3} x^{2}+5 x-2 \sqrt{3}=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=4 \sqrt{3}, b=5$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=5^{2}-4 \times...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $2 x^{2}+6 \sqrt{3} x-60=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=6 \sqrt{3}$ and $c=-60$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(6 \sqrt{3})^{2}-4 \times 2...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $\sqrt{3} x^{2}-2 \sqrt{2} x-2 \sqrt{3}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=\sqrt{3}, b=-2 \sqrt{2}$ and $c=-2 \sqrt{3}$ $\therefore$ Discriminant, $D=b^{2}-4 a...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $\sqrt{3} x^{2}+10 x-8 \sqrt{3}=0$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=\sqrt{3}, b=10$ and $c=-8 \sqrt{3}$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $\sqrt{2} x^{2}+7+5 \sqrt{2}=0$. Comparing it with $a x^{2}+b x+c=0$, we get $a=\sqrt{2}, b=7$ and $c=5 \sqrt{2}$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(7)^{2}-4 \times...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
The given equation is $2 x^{2}-2 \sqrt{2} x+1=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=-2 \sqrt{2}$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-2 \sqrt{2})^{2}-4 \times 2...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $\begin{array}{l} 15 x^{2}-28=x \\ \Rightarrow 15 x^{2}-x-28=0 \end{array}$ On comparing it with $a x^{2}+b x+c=0$, we get; $a=25, b=-1$ and $c=-28$ Discriminant $D$ is given by:...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $\begin{array}{l} 16 x^{2}+24 x+1 \\ \Rightarrow 16 x^{2}-24 x-1=0 \end{array}$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=16, b=-24$ and $c=-1$ Discriminant $D$ is given by:...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $25 x^{2}+30 x+7=0$ On comparing it with $a x^{2}+b x+x=0$, we get; $a=25, b=30$ and $c=7$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =30^{2}-4 \times 25...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula: .
The given equation is $2 x^{2}+x-4=0$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=1$ and $c=-4$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(1)^{2}-4 \times 2 \times(-4)=1+32=33>0$ So, the...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $x^{2}-6 x+4=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-6$ and $c=4$ Discriminant $D$ is given by: $ \begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-6)^{2}-4 \times 1...
Find the roots of the each of the following equations, if they exist, by applying the quadratic formula:
Given: $x^{2}-4 x-1=0$ On comparing it with $a x^{2}+b x+c=0$, we get: $a=1, b=-4$ and $c=-1$ Discriminant $D$ is given by: $\begin{array}{l} D=\left(b^{2}-4 a c\right) \\ =(-4)^{2}-4 \times 1...
Find the discriminant of the given equation:
$\begin{array}{l} 1-x=2 x^{2} \\ \Rightarrow 2 x^{2}+x-1=0 \end{array}$ Here, $\begin{array}{l} a=2 \\ b=1, \\ c=-1 \end{array}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\...
Find the discriminant of the given equation:
$\begin{array}{l} \Rightarrow 2 x^{2}-3 x+1=0 \end{array}$ Comparing it with $a x^{2}+b x+c=0$, we get $a=2, b=-3$ and $c=1$ $\therefore$ Discriminant, $D=b^{2}-4 a c=(-3)^{2}-4 \times 2 \times...
Find the discriminant of the given equation:
Here, $\begin{aligned} a &=\sqrt{3} \\ b &=2 \sqrt{2} \\ c &=-2 \sqrt{3} \end{aligned}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\ =(2 \sqrt{2})^{2}-4 \times...
Find the discriminant of the given equation:
Here, $\begin{array}{l} a=2 \\ b=-5 \sqrt{2} \\ c=4 \end{array}$ Discriminant $D$ is given by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-5 \sqrt{2})^{2}-4 \times 2 \times 4 \\ =(25 \times 2)-32 \\...
Find the discriminant of the given equation:
Here, $a=3$ $b=-2$ $c=8$ Discriminant $D$ is given by: $D=b^{2}-4 a c$ $=(-2)^{2}-4 \times 3 \times 8$ $=4-96$ $=-92$
Find the discriminant of the given equation:
Here, $\begin{array}{l} a=2 \\ b=-7 \\ c=6 \end{array}$ Discriminant $D$ is diven by: $\begin{array}{l} D=b^{2}-4 a c \\ =(-7)^{2}-4 \times 2 \times 6 \\ =49-48 \\ =1 \end{array}$
Which of the following are quadratic equation in ?
(i)
(ii)
(i) $\left(x^{2}-x+3\right)$ is a quadratic polynomial $\therefore x^{2}-x+3=0$ is a quadratic equation. (ii) Clearly, $\left(2 x^{2}+\frac{5}{2} x-\sqrt{3}\right)$ is a quadratic polynomial....