Let the length of the rod be = x + 9 cmLet the width of the rod = x cmThe diagonal = 45 cm$x^2 + (x ...

Given S = $\frac {n(n+1)}{2}$$ (i.e) \; 276 \times 2 = n^2 + n$$ n^2 + n - 552 = 0$$\implies (n - 23...

Let one part be x and the other part be 12 - x$ x^2 + (12-x)^2 = 74$$ x^2 +144 - 24x + x^2 = 74$$2x^...

let the number be x and (x + 2)$x^2 + (x + 2)^2 = 202$$x^2 + x^2 + 4x + 4 = 202$$2x^2 + 4x - 198 = 0...

$ x + y = 8 \; \implies \; y = 8 - x$$ \frac {1}{x} + \frac {1}{y} = \frac {8}{15} \; \implies \; \f...

Let one side of the rectangle be x cmThe other side of the rectangle be x + 2 cm$\text {Area of the ...

Let the number be x and x+1$ x^2 + (x + 1)^2 = 545$$x^2 + x^2 + 2x + 1 = 545$$ 2x^2 + 2x - 544 = 0$...

Let the Breadth of the room be x mLength of the room = x + 3m$ Area = x(x+3) = 70$$\implies x^2 + 3x...

Let the shortest side = x mThen the hypotenuse = 2x + 6 m$\begin{align*}\text {The third side} &...

let the number be x and (x + 2)$ x^2 + (x +2)^2 = 340$$ 2x^2 + 4x + 4 = 340$$\implies x^2 + 2x - 168...

Let the number be x and (x + 1)(x) (x +1) = 306$x^2 + x - 306 = 0$

Let the one number be x and the other number be (x + 1)x (x + 1) = 240$x^2 +x - 240 = 0$

Answer : $p=\large\frac{15}{2}$$;q=9$

Given : $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+16}$Squaring on both sides:$(x^2 -4x + 3) + (x...

Given P(x) = $ax^2+7x+b$ $P(\frac {2}{3})$ = $a(\frac{2}{3})^2+7(\frac{2}{3})+b = 0$$(\frac {4}{9})a...

Given P(x) = $2x^2+kx-6$Given P(2) = $2(2)^2+k(2)-6 = 0$$\implies 8 + 2k - 6 = 0$$\implies 2k + 2 = ...

Given P(x) = $x^2+2ax-k=0$$ P(a) = (-a)^2 + 2 (-a)a - k = 0$$ = (a)^2 - 2a^2 - k = 0$$ \implies -a^2...

$=\large\frac{1}{2-\Large\frac{1}{\large\frac{2(2-x) - 1}{2-x}}}$$ = \large\frac {1}{\large\frac {2 ...

Solution :$9x^2-9(a+b)x+(2a^2+5ab+2b^2)=0$$x= \dfrac{-b \pm \sqrt{b^2-4ac}}{2a}$$a=9, b=-9(a+b)$$c= ...

Answer : $x=\large\frac{-1}{a^2},\frac{1}{b^2} $

Answer : $(x-25)$ and $(x-\large\frac{1}{25})$

Answer : $x=4,\large\frac{-2}{9}$

Given : $12x^2+4kx+3=0$$ \text {For real & equal roots}, \; b^2 - 4ac \geq 0$$ (4k)^2 - 4 (12)(3...

Answer : $k=\large\frac{9}{4}$

Given : $x^2-4x+a=0$For Distinct roots, $b^2-4ac >0$=> $4^2 -4a>0$=> $16-4a >0$=> ...

Given: $ax^2+ax+2=0$ ; $x^2+x+b=0$When x = 1:$ a(1)^2 + a(1) + 2 = 0$$ 2a + 2 = 0 \implies a = -1$$...

Answer : $\large\frac{-1}{2}$

Given : $ax^2+2x+a=0$$ \text {To have two distinct roots} \; b^2 - 4ac > 0$$ (i.e) \; 2^2 - 4 (a)...

Answer : $x^2-2\sqrt 3x+2$

Given : $3\sqrt 3x^2+10x+\sqrt 3=0$$ \begin{align*} D & = b^2 - 4ac \\&= 10^2 - 4(3\sqrt3)(\...

Answer : All real roots$ \text {For real roots}\; {b^2 - 4ac} \geq 0$$\implies {a^2 - 4} \geq 0$$\...

Given : $x^2+4x+\lambda$$ \text { If} \; \lambda = \text {4, then the quation can be written as (x ...

Answer : $\large\frac{b^2}{4a}$

Given : $(a^2+b^2)x^2-2b(a+c)x+(b^2+c^2)=0$$ \text {Since the roots are equal}, (b^2 - 4ac = 0)$$ (i...

$(c^2-ab)x^2-2(a^2-bc)x+b^2-ac=0$$b^2 - 4ac = 0 \; \text {for equal roots}$$ [ 2 (a^2 - bc)]^2 - 4 (...

Answer : $\large\frac{c}{d}$

$(b-c)x^2+(c-a)x+(a-b)=0$$ b^2 - 4ac = 0 \; \text {for equal roots}$$ \implies (c - a)^2 - 4 (b - c)...

Answer : $k \leq -4$or $k \geq 4$

Solution :For real roots,$b^2=4ac \geq 0$=> $a=3,b=2,c=k$Substituting the values of a,b, and c=&g...

Answer : $\large\frac{-8}{5}\normalsize < k <\large\frac{8}{5}$

Answer : $k \leq \large\frac{-4}{3}$ or $k \geq \large\frac{4}{3}$

Answer : $k\leq -4$ or $k \geq 4$

Answer : $k=\large\frac{2}{3}$ or $-1$

Given: $(k+1)x^2-2(k-1)x+1=0$$b^2 - 4ac = 0 \; \text {for equal roots}$$\implies [2 (k - 1)]^2 - 4 ...

Answer : $\pm\large\frac{5}{2}$

$\sqrt{6+\sqrt{6+\sqrt{6+..}}}$$ \text {Let x} = \sqrt { 6 + x}$$ x^2 = {6 + x} \; \; \implies {x^2 ...

Answer : $\large\frac{b^2}{a} $

Solution : https://clay6.com/mpaimg/1_a1.jpgReminder is $11x+2$

Answer : $\large\frac{3}{2}$

Given: $f(x)=kx^2-3x+5$ $ \alpha + \beta = \frac{-b}{a} = - (\frac {3}{k}) = 1$$ + \frac {3}{k} = 1 ...