Question

Solve the following equations:


`1.√(5x+4)-1=2x`

`1) \ 3√(x-1 )+11=2x`

Answer 1

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

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{$\begin{gathered}\sf √(5x+4)=2x+1 \Longrightarrow 5x+4=(2x+1)^2 \end{gathered}$}`

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

`\large\displaystyle\text{$\begin{gathered}\sf x_1=1 \qquad x_2=-(3)/(4) \end{gathered}$}`

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

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{$\begin{gathered}\sf 3√(x-1)=2x-11 \Longrightarrow 9(x-1)=(2x-11)^2 \end{gathered}$}`

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

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

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