Question

Solve the following equations:
$1.\sqrt{5x+4}-1=2x$
$1) \ 3\sqrt{x-1 }+11=2x$

Answer 1

$\large\displaystyle\text{ \begin{gathered}\sf 1. \ \sqrt{5x+4} -1=2x \end{gathered} }$

The root of the equation is solved first. Afterwards, both members of the equality are squared, the powers are developed and it is solved.

     $\large\displaystyle\text{ \begin{gathered}\sf \sqrt{5x+4}=2x+1 \Longrightarrow 5x+4=(2x+1)^2 \end{gathered} }$

           $\large\displaystyle\text{ \begin{gathered}\sf 0=4x^2 +4x+1-5x-4 =4x^2-x-3 \\ &= 4\left(x^2-\dfrac{1}{4}-\dfrac{3}{4} \right)\\ &=4\left(x-1\right)\left( x+\dfrac{3}{4}\right) \end{gathered} }$

                                           $\large\displaystyle\text{ \begin{gathered}\sf x_1=1 \qquad x_2=-\dfrac{3}{4} \end{gathered} }$

$\large\displaystyle\text{ \begin{gathered}\sf 2. \ 3\sqrt{x-1}+11=2x \end{gathered} }$

The root of the equation is cleared. Then, both sides of the equality are squared, the powers are developed and it is solved by the general formula.

               $\large\displaystyle\text{ \begin{gathered}\sf 3\sqrt{x-1}=2x-11 \Longrightarrow 9(x-1)=(2x-11)^2 \end{gathered} }$

            $\large\displaystyle\text{ \begin{gathered}\sf 0=4x^2 -44x+121 -9x+9 =4x^2-53x+130 \end{gathered} }$

 

   $\large\displaystyle\text{ \begin{gathered}\sf x_{1,2}=\dfrac{53\pm \sqrt{(-53)^2-4(4)(130)}}{2(4)}=\dfrac{53\pm \sqrt{2809-2080}}{8}\\ &=\dfrac{53\pm \sqrt{729}}{8} = \dfrac{53\pm 27}{8}; \end{gathered} }$

         

                               $\large\displaystyle\text{ \begin{gathered}\sf \begin{matrix} \ \ \ \ \ \ x_{1}=\dfrac{53+ 27}{8} \qquad &\ \ \ \ \ x_2=\dfrac{53 - 27}{8}\\ x_1=\dfrac{80}{8} \qquad &\ \ x_2=\dfrac{26}{8}\ \ \ \\ x_1=10 \qquad &x_2=\dfrac{13}{4} \end{matrix} \end{gathered} }$