Question

What is the solution of the equation? Radical functions. Thank you!

Answer 1

Given

`√(2x+13)-5=x`

We can find the value of x below

Step-by-step explanation

Step 1: Solve for x

`\begin{gathered} √(2x+13)-5=x \\ Add\text{ 5 to both sides} \\ √(2x+13)-5+5=x+5 \\ √(2x+13)=x+5 \\ Take\text{ the square of both sides} \\ (√(2x+13))^2=(x+5)^2 \\ 2x+13=x^2+10x+25 \\ Move\text{ terms to one side} \\ x^2+10x+25-2x-13=0 \\ x^2+8x+12=0 \\ We\text{ will solve the above using quadratic formula} \\ \mathrm{For\:a\:quadratic\:equation\:of\:the\:form\:}ax^2+bx+c=0\mathrm{\:the\:solutions\:are\:} \\ x_(1,\:2)=(-b\pm √(b^2-4ac))/(2a) \\ \mathrm{For\:}\quad a=1,\:b=8,\:c=12 \\ x_(1,\:2)=(-8\pm √(8^2-4\cdot \:1\cdot \:12))/(2\cdot \:1) \\ x_(1,\:2)=(-8\pm \:4)/(2\cdot \:1) \\ \mathrm{Separate\:the\:solutions} \\ x_1=(-8+4)/(2\cdot \:1),\:x_2=(-8-4)/(2\cdot \:1) \\ \mathrm{The\:solutions\:to\:the\:quadratic\:equation\:are:} \\ x=-2,\:x=-6 \\ \end{gathered}`

Step 2: Verify solutions

`\begin{gathered} For\text{ x=-2} \\ √(2\left(-2\right)+13)-5=-2 \\ √(9)-5=-2 \\ 3-5=-2 \\ -2=-2;True \\ For\text{ x=-6} \\ √(2\left(-6\right)+13)-5=-6 \\ √(-12+13)-5=-6 \\ √(1)-5=-6 \\ -4\\e-6;False \end{gathered}`

Answer: x = -2