Solve:

$\frac{1}{p}+\frac{1}{q}+\frac{1}{x}=\frac{1}{x+p+q}$

Solve:

$x(x+1)+(x+2)(x+3)=42$

Solve:

$\frac{1}{x+1}-\frac{2}{x+2}=\frac{3}{x+3}-\frac{4}{x+4}$

For each equation, given below, find the value of $\text{'}\mathrm{m}\text{'}$so that the equation has equal roots. Also, find the solution of each equation :

$(m-3)x^2-4x+1=0$

For each equation, given below, find the value of $\text{'}\mathrm{m}\text{'}$so that the equation has equal roots. Also, find the solution of each equation :

$3x^2+12x+(m+7)=0$

For each equation, given below, find the value of $\text{'}\mathrm{m}\text{'}$so that the equation has equal roots. Also, find the solution of each equation :

$x^2-(m+2)x+(m+5)=0$

Without solving the following quadratic equation, find the value of $\text{'}\mathrm{p}\text{'}$for which the roots are equal.

$p{x}^2-4x+3=0$

Without solving the following quadratic equation, find the value of $\text{'}\mathrm{m}\text{'}$ for which the given equation has real and equal roots.

$x^2 + 2 (m- 1) x+ (m + 5) = 0$

Solve:

$(x + 5) (x-5) = 24$