Question

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Solve: $\sqrt{y-3}+2=\sqrt{4y+2}$

Short answer

Ans:$$\begin{gathered} \sqrt{y-3}+2=\sqrt{4y+2} \\ \Rightarrow y=\frac{5+4\sqrt{26}i}{9},\frac{5+4\sqrt{26}i}{9}(IMAGINARY\hspace{0.17em}ROOTS) \end{gathered}$$

Step-by-step solution

Step 1. Given information.

given expression$\sqrt{y-3}+2=\sqrt{4y+2}$

Step 2. Square both sides of the equation and solve it.   

$$\begin{gathered} {(\sqrt{y-3}+2)}^2={\left(\sqrt{4y+2}\right)}^2 \\ \Rightarrow y-3+4+4\sqrt{y-3}=4y+2 \\ \Rightarrow 4\sqrt{y-3}=3y+1 \end{gathered}$$

Again, square both sides of the equation and solve it. 

$$\begin{gathered} {(4\sqrt{y-3})}^2={(3y+1)}^2\Rightarrow 16(y-3)=9y^2+1+6y \\ \Rightarrow 16y-48=9y^2+1+6y \\ \Rightarrow 9y^2-10y+49=0 \end{gathered}$$

Step 4. Find Discriminant of the equation 9y2-10y+49 =0

Since, Discriminant $D=b^2-4ac$

here $a=9,b=-10,c=49$

$D=-{10}^2-4\times 9\times 49=-1664$

since Discriminant is negative the roots will be imaginary.

Step 5. Then the value of y.

$$\begin{gathered} y=\frac{-b+\sqrt{D}}{2a},\frac{-b-\sqrt{D}}{2a} \\ \Rightarrow y=\frac{-(-10)+\sqrt{-1664}}{2\times 9},\frac{-(-10)-\sqrt{-1664}}{2\times 9} \\ \Rightarrow y=\frac{5+4\sqrt{26}i}{9},\frac{5+4\sqrt{26}i}{9} \end{gathered}$$